Bash Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⢸⠅⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡿⡀⡏
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⡃
⠀⠀⠀⠀⠠⠂⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠸⢬⣄⠋
⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄⡀
⠀⠀⠀⠈⠳⠀⠀⠀⠠⠤⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠠⠖⠂⠋⠉⠉⠉⠓⠒⠠⡤⠙
⠀⠀⠀⠀⡀⠘⢿⡄⠀⠀⠀⠉⠀⢤⠀⢤⡤⠄⠐⠀⠀⠀⢀⣠⠤⠐⠒⠀⠒⠐⠂⠀⠐⠂
⠀⠀⠀⠀⠉⢀⣀⠡⣄⣀⣠⣀⣀⡀⠀⠀⡐⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠐⠂⠠⠄
⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠖⠋⢉⡂⠀⠀⠀⠀⠀⠀⠐⠲⠆
⠀⠀⠀⠀⠀⠀⡂⢀⠀⠀⠋⠀⣀⣀⡴⢚⣉⡝⠈⠉⠀⠘⠁⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⣘⣉⠤⢌⡁⣠⠔⠒⠀⠉⡉⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠰⠇⠶⠤⠤⣤⣀⣀⣤⠐⠒⠲⣂⠉⠉⠀⠀⠀⢈⠍
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠾⠷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⠋⠁⠀⠀⠀⠀⠈⠙⠻⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣀
⠀⠀⠀⠀⢀⣠⣴⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⣄⡀
⠀⠀⣠⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄
⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷
⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣽
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋⢀⣤⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣶⡄⠀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠀⣰⣿⣿⣿⡿⠛⠉⣀⣼⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠙⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⣼⣿⣿⣿⣇⣠⣴⣾⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠈⠙⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⡆⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⣄⡀⠀⠸⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢶⡾⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠈⠳⣀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠸⡀⠀⠘⠧⠀⡈⢱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⣤⣭⣉⠁⠀⠀⠀⠁⠀⡀⠀⠀⠰⢶⣶⢦⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⢰⣦⣿⣿⠁⠀⢶⠀⠈⠲⣄⠉⠑⣆⡀⣰⣿⠜⠛
⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠐⢮⣿⣿⣿⠀⢂⠈⢳⡄⢠⣟⡀⣼⣿⡃⠿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠸⣿⣿⡟⠠⣾⠀⠘⣇⠈⢿⡇⢿⣿⣿⣦⠄
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⣰⣿⣿⠟⢠⠀⣿⡇⠀⢸⠄⢈⡿⠀⢹⣿
⠀⠀⠀⠀⠀⠀⢰⠀⣰⡷⠿⠛⡋⢡⠀⢌⢸⣿⡄⢠⡏⠀⡾⠁⣴⣿⠿⠆
⠀⠀⠀⠀⠀⠀⡘⡸⠋⢠⠸⡇⣧⢹⠀⣼⣀⣷⡁⣸⣷⠀⣷⢸⡏
⠀⠀⠀⠀⢀⠄⠀⢀⢰⣸⠤⠓⠛⠉⠉⠁⠀⠤⠉⠛⣷⣾⣿⣤⠿⠿⠆
⠀⠀⠀⠀⣧⣤⣦⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣀⣼⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⣉⠉⠩⣛⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣷⣿⣶⣷⣿⣿⣿⣿⣿⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠶⠛⢛⡛⠛⠛⣛⠻⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢨⣷⣚⣽⣼⠿⠿⠾⠿⠓⣲⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣠⣞⣩⣷⣾⡟⠀⠀⠈⢀⣉⣉⣿⣿⣭⢶⣾⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⣴⡯⠄⢺⡓⣛⣯⠠⠤⠤⠤⠶⠶⢾⣛⣛⣻⣿⣿⣿⣿⣿⣿⡟⠂
⠀⠀⠀⢠⣿⡶⠞⢋⣛⣩⣿⡤⠀⠀⠀⢀⣀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⡀
⠀⠀⠀⣸⣫⣶⡾⠿⣟⣭⣽⠷⢶⡛⣻⣭⣽⣿⣿⣿⣿⣿⡿⠟⣫⡽⣾⢾⡿⢿⣄⡀
⠀⠀⠀⢸⢿⡥⣒⠭⢿⣩⡿⠾⣿⣍⣥⣶⣶⣿⣿⣿⢋⣡⣶⣟⢛⣉⢚⡬⠑⠁⠈⢿⣤⣤⣤⡀
⠀⠀⠀⠈⢻⣷⣵⡾⢋⣹⣷⣿⣿⣿⣿⣿⣿⣿⠟⣱⣿⣿⣿⣿⣿⣿⣿⠤⠁⡀⠀⣎⣘⣉⡛⣧
⠀⠀⠀⠀⠳⢿⣿⣶⠟⣵⣿⣿⣿⣿⣿⣿⡿⠁⣰⣿⣟⣿⣿⡿⣿⣭⣍⣈⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⠛⣿⣾⣿⣿⣾⣿⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⣏⣛⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣿⣿⣿⢿⡇⠀⠀⣿⣿⣿⣿⣿⣧⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣽⣺⡟⢸⠀⠀⠀⢻⣿⣿⣿⣷⣯⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⠸⡄⠀⠀⠀⠻⣿⣿⣿⣟⠉⠂⡀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⢷⠀⠀⠀⠀⠈⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢠⣄
⠀⠀⠀⠀⠀⠄⡀⠀⠘⣇⠀⠀⡆⠀⠰⠀⠀⡀⠀⠁⣄⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⡟⠀⠀⡜⠀⣱⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠸⡄⠀⢿⡀⠀⠇⠀⠀⠀⢀⠇⠀⠄⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⢤⠖⠀⢀⠄⠀⢀⡟⠱⠀⠀⠀⠀⠀⡆⣰⡀
⠀⠀⠀⠀⠘⢦⠀⠱⡀⠐⡇⠀⠃⠀⠀⠀⡈⠀⠠⠀⣠⠊⠂⠀⠰⡄⠀⠂⠀⣠⣁⡤⠆⠁⠀⢠⠎⠀⢦⡇⠀⠀⠀⠀⡃⠀⢣
⠀⠀⠀⡎⠳⣄⠑⡀⠱⡀⢹⡀⡠⠀⡆⠀⡇⢠⠃⡐⢁⠔⠇⠀⠀⠘⢷⣤⠶⢉⣁⣠⡶⠆⠊⠁⠀⠀⡸⠁⠀⠀⠀⠐⠁⠀⠀⣆
⠀⠀⠀⠈⠢⢀⡑⠈⢆⠡⠈⣇⣧⢠⠃⡐⢀⠏⢈⢔⠡⠊⠀⠀⠀⠀⠈⠫⣤⣤⣀⢀⢠⣶⣾⣖⡦⠊⠀⠀⠀⠀⠀⡴⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠉⠒⢶⣷⣁⢹⣳⡞⢀⣧⣮⡜⠞⠁⠀⠀⠀⠄⠀⠀⠀⠀⠈⠉⠉⠒⠛⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠹⢿⣷⣓⣸⠿⠣⠤⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠀⠀⠀⢀⡄⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁⡿⠠⠀⠂⠈⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠀⠇⠀⠀⠀⠀⠦⠀⡌⠀⠀⠶⢀⡄⢀⣀⠀⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⢸⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠑⠀⢸⠀⠐⠀⠀⠌⠀⠄⠀⡎⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠘⠖⣆⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠱⡀⢁⠀⠀⢸⡀⠁⠀⡴⠀⠄⢠⠏⠀⠀⠀⢠⢰⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠔⠀⣄⢀⡔⠊⡀⣹⡤⢤⠀⠀⠀⠀⠀⡈⠁⠐⠢⠴⢀⠓⠜⢸⡟⠁⠀⣲⡇⠘⠧⣠⠀⠀⠀⢸⡈⠀⠀⠀⠀⠀⠀⢠⣠⠃
⠀⠀⠀⠀⡎⠀⠀⠀⠀⣐⡤⠘⠉⣄⡼⢦⡀⠀⠀⠀⠈⠋⠀⠀⠀⠀⢀⠀⠋⠀⠀⡞⠁⠀⢠⠎⠀⠀⠀⠀⠀⠹⠉⠀⠀⢀⠷⣆⣸⡇
⠀⠀⠀⠰⠁⠀⠂⠈⠉⠀⣴⠧⠘⡟⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠈⠃⠀⠀⠘⠥⠐⣤⠂⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢧⣿⡿
⠀⠀⢀⠜⠀⢀⠀⠶⢆⠀⠀⣢⣠⠇⢰⡟⠀⠀⠀⠀⠡⠀⠀⠀⠘⡀⢠⡆⠀⠀⣆⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⢌⢿⡇
⠀⠀⢠⠀⠀⠀⠀⣀⣈⠦⠖⠁⠆⠀⣼⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠉⠛⠋⠀⠀⠄⠀⠈⠉⠐⠒⢤⣀⠀⠀⠀⠀⠘⡈⡇
⠀⠀⠀⠁⠀⠀⠐⠉⠀⠀⢀⡖⠀⢀⠟⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⢀⠀⠀⡙⠀⠀⢆⠀⠀⠀⢣⠇
⠀⠀⠀⠀⠀⠡⡀⠀⠠⠞⠡⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⠉⠙⠺⢇⠀⠐⢁
⠀⠀⠀⠀⠀⠀⠈⠙⠅⢀⠜⢆⠀⠀⢿⡲⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡞⢀⡰⠚⠓⢦⡀⠈⡃⠀⠈⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⠛⠃⠀⣼⡗⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡅⠀⠀⠀⢰⠈⡇⠸⣧⡀⢈⣽⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢠⣟⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣌⣡⡤⠞⢀⡾⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣘⠥⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠆⢀⣀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠊⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⡨⣷⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⢀⢀⡄⠀⠀⠰⡈⠈⣳⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢋⡀⠉⠒⠀⠿⠳⠺⣷⠶⠻⠞⠉⠉⣯⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡀⠀⠀⠀⠀⠀⢻⡆⠀⠎⣷⣶⣟⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡴⠶⠦⠾⠗⠲⠿⠧⠤⠦⣼⠯⠶⢶⠿⠛⠉⡀⣤⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠋⠀⡀⠀⢀⢀⢁⠀⠀⠂⢠⣄⣦⣶⠄⠄⣾⣿⣿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢀⣀⣠⡧⠤⠬⡾⣾⠦⠒⠨⠬⠿⠿⠛⠛⣛⣟⠿⠋⡼⠁⢷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠔⠊⠉⠀⠀⡀⠀⠀⢸⡆⠀⠀⠀⣼⣧⢀⠄⡿⠋⠰⢲⣷⣧⠀⠈⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣧⡀⣦⣈⣧⣣⣰⣬⠿⠿⠶⡻⠇⠤⣠⣿⣿⠥⠀⠀⠠⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣉⠀⢠⡤⠖⠋⠉⠉⠉⠉⢸⡅⡠⢰⡴⠋⠀⠀⣰⡿⠂⠂⠀⠀⢀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠁⠀⠀⢠⣀⠀⠀⠀⣀⣿⡤⠋⠀⠀⢠⣾⣿⣫⣉⡁⠀⠠⠄⠀⢀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⢠⠀⠡⣷⣽⣦⠤⢿⡟⠀⠀⠐⢡⡿⠿⢮⣍⣉⠘⠓⠢⢄⡀⢀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠷⠛⠉⠁⠀⣴⠎⠀⠀⢀⣴⣟⣿⣿⡏⠁⠀⠈⠀⠀⢀⣈⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡀⠀⣨⣼⡏⠀⠀⠀⠂⣿⡷⠦⠠⠁⠀⠀⠀⠀⠀⠡⡆⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠋⢁⢼⡇⠀⠀⠀⠅⣿⣶⠮⠀⠀⠀⠀⠀⠀⠀⠨⠄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣸⠿⣿⠀⠀⠠⢸⣿⣽⣯⣑⠢⠠⠤⡦⠌⢀⠔⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢁⣾⠁⠀⢀⣄⣿⣏⡭⣕⠪⡁⠑⡡⠀⣩⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⢹⠀⢀⢮⣿⣿⡦⡭⠂⢑⡨⠞⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢰⡞⣰⣯⣟⣛⠻⠉⣒⣉⣁⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣿⠰⡷⢋⣁⣠⠴⠊⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠛⠛⠛⣿⣿⠿⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣴⣿⡟⠋⠁⢠⣿⣿⣷⣶⣶⣦⣾⡏⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⣿⡿⠛⠋⠉⢻⣿⣿⣶⣶⣾⣿⡿⠉⠉⠉⠻⣿⣿⣧⣤⣄⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠿⠿⣿⣿⣏⣠⣤⣶⣶⣿⡿⠛⠉⠁⣸⣿⣧⣤⣀⣀⣾⡿⠋⠙⠻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣯⣤⣶⣾⣿⣿⣿⠟⠋⠉⣹⣿⣧⣤⣴⣾⣿⠟⠛⢿⣿⣿⣿⣤⣀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠟⠻⣿⣿⠟⠋⢉⣠⣿⣿⣤⣾⣿⣿⣿⡟⠉⢹⣿⡇⠀⠀⣾⣿⠟⠛⢿⣿⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁⢀⣤⣾⡿⢿⣷⣴⡿⠛⠛⢻⣿⡏⠀⠈⣿⡇⠀⠈⠉⠀⠀⢼⣿⠏⠀⠀⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣀⣶⡿⠋⠁⠀⣠⣿⣿⣷⡄⠀⠀⠻⠟⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣄⡀⢀⣾⡟⠉⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠉



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣭⣮⣯⣴⣬⣽⣬⣼⣯⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠂⢤⡆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠓⠈⠁⢀⡀
⠀⠀⠀⢀⠀⠀⠈⠀⢩⠚⣀⠀⢄⠀⠂⠰⢠⣄⠀⡘⣠⠇⢀⣀⣴⠆⠁⠈⠁⠀⠀⠻⢆⣈⠩⣼⡆
⢠⠄⢈⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⡾⠇⠀⠠⠼⢏⣁⡛⣹⢯⠍⠂⢸⢧⢀⣰⣆⣃⣴⢹⣍⢾⣧⠹⣤⡀
⠈⢰⡎⠂⠀⠀⠀⠀⠀⠀⠘⢠⠴⣎⢁⠀⢘⣤⢼⣧⣿⣿⣽⡀⣤⣬⢹⣿⠛⢹⠛⠿⠋⡘⠚⠛⡶⡹⡏
⠃⡵⠆⠀⠈⣤⠀⠐⣶⣖⣤⢽⣄⠈⢷⡂⣬⠀⠀⣹⠀⡻⣿⣽⣧⠹⡀⠤⢚⠙⠛⣃⠐⠄⠄⣀⡇⣃⢿
⣾⣄⣦⣆⡾⢖⢃⣿⣿⡏⠙⠛⠻⣞⣀⢙⢨⠁⢨⡷⡄⠹⡷⢏⠀⡀⠡⣄⣾⣄⠀⠙⡐⢀⠶⠫⣶⡿⡛
⡿⠿⣿⣿⠏⠂⣿⠋⢡⣃⡐⡀⡘⠉⣸⣆⠘⢃⠖⠑⣲⡆⠀⠀⠀⠀⠴⠁⠍⢁⣦⠔⠀⢃⢸⢶⡿⠷⠂
⠒⠲⠟⠨⡍⠉⠍⣺⡈⣉⠧⢀⣴⣟⣊⠂⠀⠖⠚⣻⡚⠛⣤⠀⢠⣤⣤⡾⣇⣴⡆⢨⡞⡇⠥⠾⢄⠑⠐
⡀⢣⠀⢰⢉⣧⡀⡶⠛⢈⡀⡿⣼⢿⡉⣭⣴⡴⠋⢉⡙⠉⠉⡦⢠⠽⡄⡷⠚⠇⢀⢉⣆⠥⣴⣦⠐⠀⠆
⡷⢻⣾⣆⣤⠈⠧⢳⣤⡾⣬⢍⢡⢈⣀⠊⡤⢀⣖⠞⣳⡶⠷⠶⠀⠠⣶⢣⠀⠐⣾⡌⠫⠶⠆⠀⢨⣿⣿
⡷⠶⠿⠏⣠⣾⣿⣿⡧⢸⣿⣟⣆⣺⣻⣞⣤⣬⡴⠞⣹⡷⠀⣄⡀⠀⢛⠀⢠⢠⣿⣿⡂⣠⡄⢠⣰⣬⣿
⣿⢋⣙⡦⠛⢿⢫⣾⣷⣾⣿⣿⣿⣿⢿⣶⣯⣭⣶⣠⣿⣷⣤⡏⢹⡅⡐⠊⢷⣜⣣⣭⣥⣾⣽⣿⣷⣽⣿
⣟⡿⣿⣿⣤⣿⣷⣨⣿⣿⣿⡿⣿⣾⣿⣿⣟⣿⣻⣟⣻⣿⣿⣿⣿⣻⣷⣠⣼⢿⣿⣿⣿⣿⣿⣽⢿⣿⣿


⠀⠀⣠⣴⠖⢀⣠⣴⣶⣶⣦⠀⠀⠀⣠⣶⣶⢶⠶⣶⠶⣶⣶⢶⣶⣄
⠀⠜⠋⣿⣿⣿⣿⣿⣿⡟⢁⠀⠀⠀⢻⣿⣿⣤⣼⣿⣶⣴⣤⣾⣿⡟
⠀⠰⣾⣿⣿⡿⠟⠛⣻⠷⠋
⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠛⠉⠉⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣗⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢻⣦⡀
⠀⠀⠀⠀⢀⣴⣟⣟⡻⠡⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠙⢦⡀
⠀⠀⠀⣴⠟⠹⣿⠜⠱⡀⡃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠡⢶⣿⣆
⠀⠀⣾⡿⢿⡾⠃⣤⠃⠉⣡⢸⡀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⢢⠀⠀⢢⠀⠀⠲⠀⠠⡀⠹⠿⣦
⠀⡼⣫⣷⡼⡁⢟⠁⢁⠂⠀⠀⠸⠡⠳⠀⢴⣄⡀⠀⣠⠀⠀⠈⠀⠁⠀⠐⠂⠀⠀⢆⠠⠀⠀⠘⠀⠁⠁⠾⠉⣧⡀
⢰⠛⡻⠚⠗⣾⡁⠀⠍⢀⢀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⡄⠀⠀⢀⠁⠐⢄⠀⠄⠰⣆⠀⢵⣿⡄
⣸⠛⢶⡘⣾⣽⣈⡘⠐⠀⠀⠀⠒⠚⠰⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠈⠀⢂⠀⢠⠰⣦⡒⠚⡈⠱⣿⣦
⣿⢤⢏⢫⣌⣝⣱⣧⣟⡀⣤⠘⠀⠀⠠⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⢂⠀⠆⠀⠀⠀⠀⡄⣄⠈⠄⢧⠀⡊⣿⠁⠁⢘⣘⢿
⢹⣶⣃⣘⣎⣩⣻⢻⢙⣵⣿⡆⡀⣥⡴⠄⢠⠀⠀⢰⠀⡀⢸⠠⠀⠈⠀⣴⣀⠀⢼⡂⠁⡹⣧⡘⢦⢶⠀⠈⠳⣄⠘⢹⣷
⢸⣿⢛⠟⣶⣇⠙⡂⢆⣼⣄⡇⢫⣿⣇⠨⡸⣏⣀⠸⣚⢳⡈⣄⣶⠂⢚⣻⣄⠐⡏⣵⠠⠏⢻⣷⠀⢉⢻⠄⢈⢹⣠⡆⡙
⠸⣷⠋⣵⢧⡙⣔⠀⡘⠾⡾⠠⣜⣿⠃⣤⣟⣮⠠⠰⠟⡛⣥⢧⣭⠂⡸⣩⣸⡀⢂⠹⣢⡠⠎⢟⡱⠀⠀⠀⣌⣹⠱⡝⣦
⠀⠈⣧⠛⣿⣻⠃⢠⣇⠀⠁⠀⢏⡺⠇⠀⢺⠏⠁⠰⢿⣅⡇⡲⢻⠔⢀⡉⣛⡄⠀⠉⢋⠃⠀⠈⠘⢡⠀⠀⢋⢧⡶⠺⢿
⠀⠀⢿⠰⣳⠃⢄⣿⣟⡜⠀⠐⠀⠘⠀⠀⠤⠄⠀⠂⠒⣖⠐⢹⠚⣫⠢⠘⢅⠼⠀⠀⠈⠐⠀⡠⣴⣿⣆⠀⠀⣿⣧⣠⠟
⠀⠀⠀⠙⠻⣄⣼⣟⣧⠃⠃⣧⢀⡌⠀⠀⡀⠀⢀⢀⠀⠁⠀⠀⡂⠀⡀⠀⠆⣀⡦⡐⢀⣻⡇⠊⠸⢻⠇⣄⠀⠙⣼⡇
⠀⠀⠀⠀⠀⢸⣿⣼⣎⠈⢘⠪⢩⠩⠁⣸⣗⡆⠀⣨⣰⡦⠶⡆⠁⣧⣹⢄⣰⣿⣡⡁⠸⣽⡷⠄⢓⠂⣝⣼⣮⠿⠞⠁
⠀⠀⠀⠀⠀⠈⠙⠺⢦⣐⡟⠂⣿⠐⢄⡦⡟⡇⠀⣺⠙⣶⣴⠀⠇⢺⢉⢨⠰⣺⣿⠅⢁⠚⣟⠂⡠⢧⣈⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣮⣅⠲⣸⢫⣎⣇⢾⣺⠶⢐⡇⢎⠰⠸⢟⢨⣂⡛⠙⢃⣿⣐⣮⣶⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣾⣷⠷⣦⣿⢠⣆⣵⣖⣸⣤⣺⡷⠧⣼⣿⡾⠉⠈⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣥⣤⣬⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣆⠈⠛⢀⡀⢠⡀⠀⣴⢻⡆⠀⠀⠀⠆⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠛⠃⠀⣶⠀⣀⣸⣟⣸⡯⠞⠁⠀⢠⣷⠀⠾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣠⡀⢲⡖⢉⣦⠉⣽⡏⣿⡗⣶⣠⡖⣸⡏⣴⢆⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⠸⡇⢸⣇⠜⣿⡇⢿⠃⠇⠿⠴⣿⠙⠷⡿⣼⠇⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢸⠇⠈⢀⣤⡁⠀⠘⠃⠀⣠⡴⡄⠀⣌⠀⢠⡀⠿⠖⣹⡿⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠒⣦⡎⣽⠇⢠⡾⢋⡀⢻⣷⠃⣰⣿⣀⣿⣇⣠⢊⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⡇⠳⣦⠻⠗⢿⣇⡼⣿⣏⢸⡇⢰⣿⠈⠫⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢇⣰⡿⠁⣶⠄⢀⡞⢹⣿⡌⠁⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠶⠘⢧⣼⠿⠁

⠀⠠⣤⢄⡄⠀⢀⣀⡄⡀⡀⠀⠀⣀⡀
⠀⠀⠘⣾⣿⣻⢿⣼⣽⣟⡷⢮⣥⠁⠙⣶⣀⣀⣀⣠⣤⠄⣀
⠀⠀⠀⠘⠿⣿⣿⣾⡟⣹⣿⣿⡿⣛⣻⣿⢿⣻⣿⣟⠿⣷⣬⡁⢢⣀⣀⣠⡴⢂⣿⢦⡀
⠀⠀⠀⠀⠀⠻⣿⡿⢷⣾⣶⣿⣿⣿⣿⣭⣽⣿⣿⠿⣷⣤⣙⠻⣷⣽⣿⣷⡆⣸⠋⠘⠻⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠈⣷⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⣄⣙⢿⣿⣿⣇⡄⠀⠀⠀⠀⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣧⣄⣀⠀⠀⠀⠀⠉⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄⡀⠈⠙⢿⣿⣿⣿⣿⣿⣶⣎⠀⠀⠀⡈⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠻⣿⣿⣿⣿⣿⣿⡛⠻⠿⣿⣿⣷⣦⣤⣛⠿⣿⣿⣿⣿⣿⣶⣶⡊⢀⠄⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣯⣿⣿⣿⣿⣿⣿⣿⡟⢽⣿⣿⣿⣷⣥⣠⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡿⠿⢿⢿⣽⣶⣿⣿⣿⡿⣷⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣷⣯⣿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣷⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⣿⣷⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠀⢈⡙⢻⢿⣦⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠔⠒⠛⠃⠦⠤⠭⠯⢩⡍⠉⠤⠭⢛⢾⣷⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⡟⠣⠤⣬⠤⠠⠥⠬⢀⡦⠤⠄⠌⣵⣶⣿⠴⠦⠿⣿⣿⣿⣿⣿⣳⣛⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣠⡋⠀⠤⠄⠀⡤⠠⠠⡀⠠⠤⠤⠤⠤⠤⠴⣶⣶⣶⣶⣟⣿⣿⣿⣿⣿⢿⣯⣙⠄⠰⠧⠿⢦⣄
⠀⠀⠀⠀⠀⣰⠥⠀⠀⠉⠉⢹⠉⠉⣉⣉⠉⠉⣙⠉⠙⠉⠛⠛⣉⣉⣉⣩⣴⣿⣱⣿⣟⣿⣿⣿⣿⣶⣦⣀⠈⠻⣷⣆
⠀⠀⠀⠀⠸⠃⢀⠠⠩⠙⠷⠾⠖⠲⠶⠤⠧⠭⢿⡭⠭⠭⠭⠭⠿⠿⠛⠛⣛⣉⣿⣿⢏⣿⣿⣿⣿⣿⣿⣿⠛⠆⠀⠙⢷⡄
⠀⠀⠀⠀⠃⠀⡀⠠⠀⠰⠒⢲⣖⢒⣒⣒⡒⣒⡒⣷⠒⠒⠒⠲⣖⣿⣻⣻⣽⣿⣿⣟⣩⣿⣿⣯⣽⣿⣯⣭⣇⣀⠀⠀⢸⣿⣷
⠀⠀⠀⢰⡂⠄⠐⡆⠉⢉⣀⣈⣭⣁⣀⣈⣅⣉⣈⣉⣉⣉⣉⣉⣉⣥⣤⣤⣼⣾⣿⡯⠴⣿⣿⠉⣈⣙⣿⣗⠏⠀⠀⠀⠁⠀⠹⣇
⠀⠀⠀⠈⠀⠤⣚⣯⡭⠤⠴⠖⢶⡖⠖⠶⠶⠶⠶⢶⠖⠒⠲⠶⠒⠟⢛⣛⣹⣭⡷⢖⣿⣿⣿⣿⣧⣽⣯⣿⣉⠑⠒⠀⠄⠁⠀⠘⣆
⠀⠀⠀⢈⡀⠈⠀⢛⣢⠠⠦⠰⠈⢥⠀⠀⠐⠐⠒⠊⡓⠀⠀⢘⣉⣩⣭⡽⠿⢟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣭⡴⢨⠍⠐⠢⢄⣀⠈⡜
⠀⠀⠀⠀⠄⣦⠀⠉⠷⠀⠄⠀⠐⠀⡄⠀⠀⢀⣀⣘⣆⣁⣨⡬⠷⣴⡾⢏⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⠤⣀⠀⠀⠠⢄⡙⢘
⠀⠀⠀⠀⢤⠘⠀⠋⠁⣭⡤⠀⠒⠂⣙⣀⣁⠠⠤⠤⢦⣆⣖⣺⡿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣽⣟⡛⠛⠷⢦⣍⡂⢀⡀⠈⠹⠡
⠀⠀⠀⠀⠀⡁⢀⠑⢀⣠⠵⠒⠉⣉⣀⣤⡤⠶⢖⣚⣋⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣅⣘⠐⠂⢌⡙⠳⣤⣑⠂⠢
⠀⠀⠀⠀⠀⠘⢤⠟⢁⣠⠶⠞⢁⣩⠴⠖⣒⣯⢭⣶⣾⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿⣏⡉⠉⠛⠻⠶⣤⣌⠓⢬⡙⠻⠟
⠀⠀⠀⠀⠀⠀⠈⢦⡟⠁⡤⠞⢉⡤⢞⣯⣵⣾⣿⣿⠿⠛⢉⡴⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆⠀⡁⠒⢤⣀⠈⠙⠻⣶⣌⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢾⢀⡴⢩⠞⣩⢞⡭⠋⠉⠀⠀⣰⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠀⠉⣶⣮⣙⠶⣄⡈⠛⠳⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⡏⣴⣣⠋⠘⣀⠀⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠳⢦⣄⡀⠀⡉⠻⢷⣤⡙⠶⣤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⠈⠀⡀⢊⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠓⢦⣄⠀⠈⠛⢿⣷⣤⡁⠀⠙⠻⢷⣄⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠂⠠⢰⡌⠻⣿⡏⣿⣿⣿⣿⣷⣯⣭⣛⡻⠛⠀⠀⠈⠁⠘⠒⣄⡀⠀⠙⠿⣷⣤⡀⠈⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⣿⣷⡄⢹⢇⣿⣿⣿⣿⣿⣿⣿⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⣄⡀⠈⠛⢻⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠁⠔⢸⣿⣿⠘⢸⣿⣿⣿⣿⣿⣿⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠙⢶⣄⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡇⢸⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⠋⠛⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⡐⢋⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢸⣿⣿⣿⣿⣿⣿⣿⣟⠦⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠳⣄⡑⠫⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠉⠻⣿⣿⣿⣿⡛⣿⠳⡌⠱⣄⠀⠱⣆⠀⠐⣄⠀⠈⢳⣄⠀⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢹⣿⣿⣦⡹⣎⢳⡜⢆⠈⢣⡀⠘⢷⡀⠈⢧⡄⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡼⠀⠉⠻⣷⠸⡆⢳⡈⢆⠀⢷⠀⠈⣷⡀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠃⣹⡈⡇⠘⣇⢘⡇⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡀⣀⠀⠀⢠⡆⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠀⠷⢤⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⢰⡀⠀⠲⡆⠎⠀⣴⠀⢀⡠⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⠈⠀⠁⠀⠀⠏⠀⠐⠁⠀⣹⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡾⢦⠀⠀⣠⠤⠶⢤⣤⣤⡴⠶⢦⣄⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡜⠁⢰⡞⠋⠉⠀⠀⠀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠐⠶⢀⡷⠀⢿⠀⠀⠀⢱⣤⣧⠄⠀⠀⠀⠙⢷⣤⣤⣴⣶⣶⡶⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡇⠀⠈⡏⠑⠀⣸⣿⠉⠁⠀⠀⠀⠀⠘⣿⣍⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠂⠀⣹⠖⠋⠻⠦⠚⠛⢻⡄⠀⠀⠀⠀⠀⣠⣿⣯⣭⣤⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢿⡄⠀⢀⣀⣀⣄⣠⣿⠶⠤⠴⠀⢀⣽⣿⣿⣯⣭⣭⣉⣉⡉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡤⢰⡏⠉⠀⠀⠀⠀⣹⣿⠄⠀⠀⠀⠀⠹⣿⣟⡛⠛⠛⠛⠛⠛⠛⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢸⠄⠈⠳⣄⠀⢤⡤⠖⠋⠀⠀⠀⠀⢀⣠⣾⡿⢿⣷⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠄⣀⠈⠓⠶⠶⠤⠤⠶⠶⠖⠛⠛⠉⢉⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠿⠶⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡌⠀⠀⡀⠀⠀⠀⣰⠀⠀⣄⠀⠀⢳⡤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡁⠇⠀⠠⠇⢠⠀⠘⠆⠀⠙⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠁⠠⠀⠀⢰⠚⠃⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡆⠉⠀⠲⣤⣀⣀⣀⡤⠠⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡏⠀⠀⠀⠘⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣰⠟⠎⠀⠀⠀⠀⠀⠹⣿⣿⣿⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠱⡻⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⢻⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠪⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠙⢾⣿⡻⣿⣯⡻⣿⣶⣄⣀
⠀⠀⠀⠀⣐⠩⠑⡠⢄⠀⠀⡠⡪⠀⠠⢀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⢻⣿⣮⡛⢿⣷⣝⡻⣿⣷
⠀⠀⠀⠀⠀⢀⠠⠀⣀⡴⡫⠞⢁⡄⠀⢁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢂⠢⡈⠻⣿⣦⡛⠿⣿⣷⣬⣓
⠀⠀⠀⠐⠈⢁⣴⣾⡧⠀⢀⡴⠋⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠑⣌⢦⡀⠻⣿⣦⣌⠛⢿⣿⡇
⠀⠀⠀⢴⣾⡿⠟⠁⢀⣴⡿⠑⠀⣠⠃⠀⠀⡀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⣄⠈⠻⣽⣦⡈⠙⢿⣷⣦⣽⣿
⠀⠀⠀⠿⠋⠀⣀⣴⣿⡿⠀⠀⡔⢁⠀⠀⡀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀⢀⠀⢧⣉⣦⣄⢨⣻⣿⣦⣤⣹⣿⣿⡁
⠀⠀⠀⢠⣴⣾⣿⡿⠋⠀⢠⠞⡠⠀⠀⣼⠁⠈⠀⡄⠀⠀⠠⢀⠀⠀⠀⠀⠀⠸⢀⠇⠀⠠⠀⠄⠈⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠙⡿⠋⠀⠀⣴⠋⡐⠁⢀⡜⠁⠀⠀⣸⠁⠀⠀⠀⢸⠀⠀⠃⠀⠀⠀⠃⠁⠀⠀⢃⠘⠂⠀⠻⣿⣙⣿⡛⠛⠛⢿⣿⠉
⠀⠀⠀⠀⠀⠁⠀⣠⡾⣡⡞⠀⢀⡾⠁⠀⠀⢠⠃⠀⠀⠀⡀⣸⠀⠀⢱⠀⠀⠀⢸⣶⠄⠀⠈⢄⠀⠄⠀⠘⢿⣹⣾⣧⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠻⣴⠋⠀⢠⡿⠉⡀⠀⠀⡎⢀⠀⠀⢀⡇⡼⠀⠀⢨⠀⢃⠀⠈⣿⡇⠀⠀⠘⣿⣆⠀⠀⠠⠻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⣿⣇⡴⠁⠀⣸⠃⡂⠀⠀⢸⠇⡧⠀⠀⠨⡇⢈⠀⠀⢸⣿⡦⠀⠀⠈⣿⣧⡀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠁⠀⣰⣿⢐⠁⠀⠀⣿⠢⡇⠀⠀⢠⣿⠐⠀⠀⠀⠻⣿⣄⠀⠀⠘⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠋⠠⠄⣰⠯⢸⠇⠀⠀⢸⣿⣧⣇⠀⠀⠐⠍⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣿⣿⣭⣿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⢿⡭⠿⠿⢿⡭⠭⠿⢷⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⢿⠉⢉⢻⡿⠿⠟⠛⠛⠛⠛⠻⢿⣟⣣⣬⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣽⣿⣛⠛⠿⠿⣿⣷⡶⠒⠒⠒⠒⠾⣿⢟⣿⣁⣬⣽⣷⡿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⣠⣞⡉⠉⠉⠛⠿⠛⠀⠀⠀⠒⠲⣾⠛⠛⠛⠉⠈⠁⠨⠟⢦⣴⡟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠆⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⠀⠀⠀⠀⠀⠀⡰⣟⠩⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⡀⠀⠀⢈⣿⠷⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣧⣄⠀⠀⢀⣤⣷⣿⠀⠈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣼⡷⠂⠀⠘⠋⠀⠀⠀⠠⠀⠀⢨⣷⣤⣀⣀⠀⠙⠃⠀⠁⠀⢨⣿⢠⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡷⠀⠀⠂⠀⠀⠀⠀⢀⣤⠴⠶⢷⡟⠋⠉⠀⠀⠀⠀⣶⣶⣾⣿⣿⠀⠀⠠⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠁⠀⠀⠀⠠⣴⣶⣶⣾⣿⣶⡿⠛⠀⠀⠀⠀⢀⣤⣶⢿⣿⣿⢣⡟⢠⣶⡦⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠿⠉⠼⠿⢿⣥⣀⣀⡀⠀⠀⠀⠛⠛⢙⣿⣿⡿⣿⠃⠤⠈⠐⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠑⢀⣤⣀⣀⠀⠀⠀⠀⠘⠋⢉⣉⣉⠉⠉⠛⣿⠿⣂⠀⣉⡁⣿⠀⢀⣴⡶⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡎⠀⠀⠙⠉⢢⣄⣀⠀⠀⠀⠀⠈⣿⡷⡶⠀⣼⣿⠛⠀⣡⠀⣿⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡆⢀⠀⠀⠀⣫⣴⡂⠀⠀⠀⠀⠘⢁⣴⣾⣿⡓⠚⠋⡿⣶⣿⠐⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡁⠀⠀⠀⠀⣿⠇⢙⣤⠀⢰⣶⠚⠁⠉⠙⠛⠋⣤⡿⠀⣿⣿⠐⢾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣶⣄⠀⠀⠀⠀⠘⠃⠀⣼⡟⠀⠀⠀⠀⢠⣾⣿⡇⢠⣿⡿⢸⣿⣿⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢉⣿⠋⠀⠀⠀⠀⠀⠀⠈⢥⠀⠀⠀⣠⣿⣿⡿⠀⣾⣿⡇⢈⣵⠟⢩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⡏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣴⣿⠿⠛⢃⣠⣿⣿⣿⣴⠛⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⣾⡿⢦⣀⣠⣄⠈⠙⠁⣴⢶⣾⣿⣿⠓⣿⡏⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣤⠀⠀⢻⡿⠁⠀⠙⣿⡿⠁⠀⣉⣷⣾⣟⣿⡟⠀⡿⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣥⣤⣄⡼⠇⢰⣶⣾⣿⣗⠀⠚⠻⣿⣿⣿⠃⠀⢠⡗⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⢻⣏⠀⠀⠸⢿⣿⣿⣿⠃⠀⠀⠋⣿⠈⠄⠀⢸⡇⢀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡟⠛⠲⣾⢿⣾⣿⣿⡏⠀⠴⢁⡾⠁⠀⠀⠀⣸⡷⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⣠⡄⠀⠀⠈⠛⠋⠁⢠⢲⡟⣥⠀⠀⠀⠀⣿⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠃⠰⠶⠀⠀⠀⣀⠈⣾⠹⠀⠀⠀⠀⢸⡟⠑⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⠀⠘⡁⠞⢹⡏⠀⠀⠀⠀⠀⣾⠄⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠾⠃⠀⣀⡄⠀⠀⣼⠃⠀⠀⠀⠀⠀⣿⡗⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⠈⠠⠈⠘⡿⠁⡠⠃⠀⠀⠀⣿⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡃⠀⠄⢁⡴⠒⠄⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡆⢠⠇⠀⣀⣠⣄⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡃⣜⣠⠟⠉⠀⠉⠙⢦⣄⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⣶⣿⡟⠳⣶⣶⣿⣿⡿⢿⣶⣶⣶⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡟⠉⣿⣿⣿⣿⠃⢱⠟⣻⣿⣿⠁⠈⢻⣷⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⢸⡏⣿⣿⡇⠀⡀⢸⣿⡇⠀⠀⠀⣿⣿⠃⠀⠀⠀⣼⣿⡿⠀⢨⣿⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿⣿⢿⣿⣿⣿⡉⠈⡇⢿⣿⡇⠀⠃⢸⣿⡁⠀⠀⢰⣿⡿⠀⠀⠀⣸⣿⡟⠀⠀⠈⣿⣿⠟⠉⢹⣆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣏⡄⠁⢿⣿⣧⡀⠀⢸⣿⡿⠀⠀⢸⣿⠛⢶⠀⢸⣿⠇⡄⠀⢠⣿⡏⠀⠀⠀⣼⣿⠃⠀⠀⢾⣿⣦⣤⡀
⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⢿⣹⣿⣿⣄⠀⠘⣿⣿⣇⠁⠘⣿⡇⠀⡆⢸⣿⠀⣢⠀⣿⣿⠀⠐⠀⣼⡿⠛⡄⠀⣸⡿⠃⠀⠀⢀⡞⠛⠛⠉⣿⡀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣵⣷⠈⢻⣿⣟⡁⠀⢿⣿⠉⠙⠀⣿⡇⠀⡇⢸⡿⠀⣧⠀⣿⡿⣄⠀⢰⣿⡇⠀⠇⣰⣿⡇⠀⠀⣰⣿⡿⠂⠀⠀⡿⣇⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⢻⣿⣿⠧⡀⢿⣿⡁⢀⠸⣿⡿⠻⣧⣿⣇⠀⣿⣾⣿⠀⡿⢸⣿⠁⢹⡇⣼⣿⣦⡄⢰⠟⠃⡁⠀⣴⣿⡟⠀⠀⠀⠀⠀⣿⡿⢷⡄
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣀⣹⣿⣶⠿⣾⣿⣿⠛⢷⣿⡷⠀⠅⢹⣿⠀⡗⢸⣯⢀⡇⢸⡿⠀⠀⢱⣿⠁⢹⠃⠀⠀⠀⢡⣾⡿⠋⠁⠀⠀⠀⣠⡶⠋⠁⠀⣷
⠀⠀⣿⣿⣿⣿⣟⠻⣿⣿⣿⣍⣻⣿⣿⣿⣿⣿⠀⠀⢹⣿⠀⣤⢸⣿⠀⢧⣼⡏⢸⡇⣾⡇⠀⠀⣼⡏⢠⡍⣶⠶⠀⠀⣼⣿⠂⠀⠀⢀⠀⠿⠃⠀⠀⠀⠨⣟
⢀⣴⣿⣿⣿⣿⣟⣧⣼⣿⣿⣿⣷⣿⣿⠀⠹⣿⣷⠀⢾⣿⡆⠘⢿⣿⠀⠈⣿⡇⠸⠁⣿⠁⢰⣷⣿⠁⢶⣴⡟⢀⠙⣾⡟⢻⡆⠀⣠⡿⠗⠀⠀⠀⢀⠀⠒⡿⢷
⢸⣷⣿⣿⣿⣿⠿⣿⣿⣴⣿⣿⣷⡿⢿⣶⠖⢿⣿⡆⠘⣿⡇⠀⠸⣿⠀⠀⣿⡇⠰⢠⣿⠀⠘⣿⡏⠀⣹⣿⠁⣾⣾⡏⠀⠞⢠⣶⡏⠀⠀⠀⢀⡀⠙⠆⠀⠀⢸
⢸⣿⣿⣿⣿⣷⣿⣿⢿⣿⣿⣿⣿⡶⠘⢿⣿⠺⣿⣇⠀⢹⣿⠀⠀⣿⠀⠀⣿⠃⠀⢸⡇⠀⢰⣿⠃⠀⣿⠃⢰⣿⡏⢀⠎⣠⡟⠁⠠⠀⣀⡲⠟⠁⠀⠀⠀⠀⢿
⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠸⣿⡗⢺⣿⡀⠘⣿⠀⠀⣿⡀⠀⣿⡀⠀⣼⡇⠀⣸⡇⠀⢸⡟⠀⣺⡏⠀⣜⣾⠋⠀⠐⣡⣶⠛⠁⠀⠀⠀⠀⣀⣤⢼
⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⣿⣇⠀⢿⣇⠀⣿⡆⠀⢿⡇⠀⣿⡇⠀⣿⠁⠀⣿⠁⢀⣿⠀⣸⡟⠀⠂⡝⠁⠀⢀⠶⠁⠀⠉⠀⠀⡀⠆⠀⠋⠀⢈
⢹⣿⣿⡿⢿⣿⣿⣿⡿⣿⣿⣻⣿⣏⠻⣧⠀⣿⡇⠘⣿⡀⢹⣧⠀⢾⡇⠀⣿⡇⢰⣿⠀⢾⣿⠀⣼⠃⢰⡿⠀⢄⠁⠀⠐⣤⠉⠀⠂⠀⣦⠀⠈⠁⠀⠀⠀⢀⣳
⠘⢿⣿⣿⡿⣿⣟⣿⣿⣯⠻⣿⣟⢿⣦⢿⣷⣘⣿⡟⢿⣇⠘⣿⡀⢸⣧⠀⣿⡄⢸⣿⠀⢸⡇⢸⡟⢀⣿⠁⡜⠂⠀⠀⠘⢃⡄⠀⠨⠓⠎⠀⠀⢀⡄⠀⠘⣫⠏
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣛⢿⣯⣿⣦⣻⣯⣿⣿⢻⣷⠘⣿⡄⣿⡇⢰⣿⠃⣾⡇⢸⡇⠀⣿⠁⣾⠃⣸⠇⡸⠁⢀⠀⠠⠀⠀⠀⠀⢀⠀⢿⠀⠉⠀⠀⠤⠾⠋
⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣝⣷⣜⢿⡿⣯⣿⣏⣿⡆⢻⣧⠸⣇⠀⣿⠀⢼⠃⣸⡇⢰⡿⢰⠋⣸⠏⢸⠁⠀⠂⠀⠀⠀⠀⠘⠀⣈⡴⠇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣇⠙⠿⣿⣿⣿⣿⡿⢿⡌⣿⡀⣿⡄⣿⡄⣼⡇⣿⡄⣸⠃⣾⢀⡟⢰⠂⠀⠤⢀⠰⠀⠀⣀⡾⠿⠛⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⡙⢹⣿⣿⣿⡼⣿⣿⣧⢹⡇⢿⣄⣿⡇⣿⠀⣿⢰⡏⣸⢡⡁⠀⠀⣠⠋⠀⢁⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣞⣿⣿⣿⣿⣿⣿⠘⣧⢸⣿⣽⣿⡟⢸⡏⣿⣿⡟⣾⡡⣾⡖⢀⣤⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢏⣿⣿⣿⣿⣿⣿⣿⣿⣸⣦⢿⣨⡏⢹⣿⣧⣼⡿⣿⣿⠰⠣⡿⢋⣴⣿⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣶⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣈⢻⣇⣼⣟⠘⠟⠁⢀⣠⣾⡿⡧⠀⢚⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣷⣠⣿⣿⣿⣟⢿⣿⣿⢻⣿⢿⠋⣧⣿⡶⠀⣶⡟⠁⢈⣷⣿⣀⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠈⠻⢧⡼⣿⡶⢤⠿⠃⠀⠉⠉⠙⠛⠛⠛⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡟⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠉⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠛⠛⠛⢻⣿⣿⣴⣿⡗⣤⣤⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠀⠀⠀⠀⠀⢨⠅⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠀⠘⠁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣶⣶⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⢹⣿⣿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⡔⣶⣿⣷⡆⠀⠀⠀⠀⠀⠀⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣙⣛⡁
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠛⠀⠉⠉⠉⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠈⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⢰⣦⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠇



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠐⠒⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠰⠶⠶⠶⠶⠶⠶⠆⠀⠘⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⠶⠶⠶⠶⠶⠶⠆⠀⠀⡇⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⢰⣤⡄⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠧⠤⠤⠀⠀⡃⠀⠀⠀⢀⡠⢴⣿⣶⣶⣿⣿⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⡠⠄⠒⡦⠤⠤⠴⠶⠷⠶⠶⠶⠆⠀⢀⠇⠀⠀⠀⣇⠠⠄⠼⡞⠉⠉⠙⢻⣽⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⢻⠋⢉⣉⣉⣙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⣩⣷⡶⠶⠿⣯⡉⠑⡆⠀⣿⣿⣤
⠀⠀⠀⠀⠀⢰⣶⣴⣶⡿⣧⣼⣀⣈⣉⣉⣉⣉⣑⣢⣤⣀⣀⣠⣤⣄⡀⠀⢣⣀⣀⣛⣹⠒⠀⠉⢙⣁⣀⣧⣼⣿⠏⠉
⠀⠀⠀⢀⣀⣾⣷⠟⠉⠉⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠇⠀⠀⠈⡀⠀⠀⡌⡀⠀⡿⠻⣿
⠀⠀⠀⠘⢿⣿⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡇⠀⢃⠃⠀⡇
⠀⠀⠀⠀⢈⣿⣻⣄⣀⣀⣠⣾⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⣼⠋⠀⠈⢳⡄⠘⢸⠀⢸⢸⠀⡇
⠀⠀⠀⠀⠙⠋⠛⢿⣯⡿⠿⢿⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠙⣦⡈⡇⠈⡎⡄⡃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢸⠀⠀⠀⠀⢾⡁⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢈⣧⢧⠀⢳⢃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⣠⠟⠘⡸⠀⠸⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠅⠆⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⠉⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⢢⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻


⣏⠉⣹⣛⣛⣛⣛⣛⣟⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣿⣟⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠐⡲⠤⢄⣀
⠀⠀⠀⠀⠀⢀⠄⢰⡎⠉⠉⢳⠄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⢠⠎⠀⠀⠀⠈⠓⡀
⠀⠀⠀⠠⠂⡇⠀⠈⡇⠀⠀⡸⠀⠀⡟⠐⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣎⣁⣀⣠⠿⠒⢶⠢⣄⣀⠀⠀⠀⡜⠰⡆⠀⠀⡜⠀⣀⣀⣤⠒⠈⠁
⠀⠀⢠⡆⠀⢰⠀⠀⠇⠀⠀⡇⠀⢀⠃⠀⣧⠀⠀⠀⠀⡠⠖⠲⢧⣀⣀⣀⣀⣀⣀⣠⠞⠀⢈⠈⣆⠀⢠⠃⠀⠙⢄⠀⣽⠋⡩⢀⡤⠛⠢⡄
⠀⠀⡀⠈⢄⠀⢧⠀⢸⠀⠰⠃⢀⠎⠀⡔⠁⠁⠀⢰⡛⠓⠦⢤⣀⣀⣀⠈⢉⣀⣀⣀⣤⠴⠃⣠⠛⡄⠘⣄⠀⠀⠀⠈⣇⠹⣝⡇⢱⠒⠒⢣
⠀⠀⣹⢄⡈⠣⡈⢢⠀⡄⢰⠀⠌⠠⠊⢀⡴⡇⠀⠸⢟⠲⠤⢤⣀⣉⣉⡉⠉⠉⢁⣀⣠⡤⠞⠁⡼⠀⠀⢻⡁⠀⠀⠀⠙⡧⠤⡵⠛⢢⠀⡸
⠀⠀⠘⠢⣍⠂⢌⣊⣣⡀⠀⡎⠜⠁⢂⡡⠒⠁⠀⠀⠀⠙⠲⠤⣄⣀⡉⠉⠉⠉⠉⠁⣀⣀⡤⠞⠁⠀⠀⠀⠑⠤⡤⠀⠊⠀⠀⡅⠀⢀⠷⠁
⠀⠀⠀⠀⠘⢄⣹⠿⣟⠓⢛⡹⢿⣁⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠄⠚⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠄⠀⢒⠀⢠⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⡞⠀⠈⠂⣤⢲⣀⠀⠀⠀⠀⠀⠀⣠⠊⠑⢄⡨⠲⡂⠃⢠⠇⠀⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢹⣦⠀⡶⠲⣢⠆
⠀⠀⠀⠀⠀⠀⣰⠋⢅⡠⢿⠦⣀⣀⡿⢠⣏⣱⢤⠔⡆⠀⣌⠀⠐⡶⠛⠀⣄⡨⠄⣠⠀⢹⡀⠘⠆⠀⠀⠀⠀⠀⡠⠂⣻⡀⢸⢹⠀⠧⠀⢸
⠀⠀⠀⠀⠀⢰⢇⢀⡾⠀⢸⠀⠀⠉⡁⠠⣆⠸⣤⠔⠁⣌⠀⠁⢸⠂⠀⠾⠁⡤⢢⠀⡳⠈⡇⠀⢄⠀⠀⠀⠀⢠⡇⡏⠀⠳⡈⠦⡓⠄⠀⣎
⢀⣀⣀⣀⡴⣋⡤⠏⢰⠀⢸⡀⢀⡴⡷⠉⢱⠜⠁⠀⢠⠉⠐⠂⢹⡤⠂⢣⡀⠳⠄⡴⠋⣸⠅⢀⡌⠀⠀⠀⠀⡼⡸⠁⠀⠀⠈⢦⣀⠀⠀⠈⣣
⠘⠿⣿⣿⡿⠿⣗⡻⠦⢷⡤⢷⡋⠀⢘⣄⡏⠀⠀⠀⡏⠑⠂⠐⠊⠳⣤⠇⠱⠠⢦⠀⠛⠁⣀⡼⠀⡀⠀⣀⣨⣡⡴⠦⣤⠀⠐⠋⡠⠟⣰⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠒⠚⠳⣄⣼⠀⠀⠀⠀⠀⠣⣀⣠⠴⠖⠦⠌⠶⠴⢄⣀⠰⠄⠒⠁⠀⠀⠉⠉⠉⠉⠀⠀⠉⠚⠭⢄⣉⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⢸⡄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢳⣷⡿⢞⠛⠛⠲⢾⣥⡴⠀⠀⠀⠀⠀⠀⢀⣐⡦⠚⠚⠓⠓⠴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢔⠀⢲⡋⢳⠖⠦⡄
⠀⠀⠀⠀⠀⠀⡾⠐⠀⢠⠡⠀⠀⣀⠿⣤⢤⡄⠀⠀⠀⠀⣠⠏⡔⠀⠀⠀⠈⠲⡈⢷⡄⠀⠀⠀⠀⠀⠀⠀⡔⠳⢄⠈⢧⠀⡇⢸⡇⠀⢹⢦
⠀⠀⠀⠀⢀⣠⢿⡁⠀⠘⢶⢶⣁⠀⠀⢻⢃⡄⠀⠀⣀⣴⠛⠛⢯⠉⠙⢦⡤⢤⣷⠀⠻⣄⡀⢀⣤⠀⠀⡜⠒⢄⠈⢣⠸⡆⣿⢸⠇⠀⡏⢸⣆
⠀⠀⢠⠏⠀⠀⠀⢙⠦⠤⡄⠊⠓⣄⢀⡿⣅⠀⠀⠱⡏⢉⢦⠀⠘⣆⠀⠈⢳⡀⠈⠳⡦⠛⠻⣭⡇⠀⠈⡧⣄⠈⢣⠈⡇⡇⡿⢸⠀⡼⠀⡞⢸
⠀⠀⠘⠗⠲⣄⡴⠉⠀⠀⡼⠀⠀⣈⠿⠓⠪⠷⠀⠀⣇⠀⠬⣇⠈⠻⡄⠀⠌⠷⡀⠀⠙⣦⠀⠘⠁⠀⠀⢇⠀⠱⡌⣇⢹⣧⣇⢇⠜⡀⠊⣠⠋
⠀⠀⠀⠀⠀⢀⡿⢶⢦⣤⣷⡖⣿⠭⠉⠃⠀⠀⠀⠀⠈⢦⡐⢾⡄⠉⢻⣆⡀⢠⡿⠦⠈⠁⠀⠀⠀⠀⠀⠀⠁⠈⠹⣾⣿⣿⣿⡇⢈⠠⠊⠁
⠀⠀⠀⠀⠀⠈⠀⢠⡼⠀⠁⠙⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠈⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠉⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠀⠀⠀⠀⢀⡀⠒⢠⠀⠀⠘⢻⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠈⠁⠘⢆⠀⠀⠀⠀⢀⠊⠑⠀⠀⣁⣁⣀⠎⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣂⡖⢁⠀⢀⠈⢦⠞⡩⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠰⠀⠀⠀⠀⠨⠀⠙⢌⣁⠨⠤⣸⠅⡾⡋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠈⠁⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⢀⣎⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀⡀⠔⠁⠀⠈
⠀⠀⠀⠀⠀⠀⢘⣀⠀⠀⠄⠀⠈⠁⢰⠀⠀⠀⠠⠝⢂⡀⣤⣶⠑⠀⠀⠀⠀⠀⠀⠐⠀⠐⢡⢺⠕
⠀⠀⠀⠀⡀⣀⠈⠀⠀⠐⠀⠀⠀⠄⣰⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦
⠀⠀⠀⠀⢄⠄⠀⠐⠀⠤⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠉⠑⠁⠄
⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠀⠀⣠⡎⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠓⢓⠂⠈⠙⢀⡀⢤⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡞⢀⠀⠀⠃⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⣮⢡⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢤⠌⣐⠃⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠉⠉⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠡⡀⠀⠀⠀⠠⠂⠀⠘⡩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠡⠀⠀⠀⠈⡀⠀⢀⠃⠆⠀⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⣳⠀⠀⠀⠀⠘⠀⠀⠤⠐⠉⠀⢀⠀⠀⠀⢡⠀⠘⠀⡴⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⣈⢁⠉⠥⡤⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠈⠀⢸⡀⡴⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠤⠦⡜⢡⠀⠀⠀⢘⠀⠀⠀⠀⠐⠒⠩⠀⠀⠂⠀⠀⠘⠧⠶⢀⡦⠀⠀⠀⠐⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡳⢀⠀⠈⠀⠀⠀⢀⢀⠀⠀⠤⠀⠂⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⡼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣭⡿⠿⡟⠙⡛⠟⣦⡐⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠠⠤⠀⠀⠀⢈⢡⡈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔
⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠁⠀⠀⢉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡠⠂⣠⠏⠀⠑⣤⠀⠀⠀⠀⠀⠘⠀⠐⠒⠒⠘⠒⢆
⠀⠀⠀⠀⡀⢀⠗⠀⠂⠐⠀⠀⠀⠀⠐⠀⢂⠑⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⡀⠀⠀⠄⢁⠜⠁⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⢸⡵⢉⡶⠃⣤⡀⡀⠀⠀⢐⠀⢀⡀⢒⡀⢶⡙⠦⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⢠⡞⠀⡏⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠄⢰
⠀⠀⠀⠈⢁⠟⠀⠐⠀⠨⠉⢙⠉⠉⡅⠀⢀⠀⠱⣄⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡆⢔⢸⠀⠀⠀⠀⡄⢀⠂⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠌⡠⠂
⠀⠀⠀⠀⠩⣄⣸⠃⠀⡐⠀⢸⠀⠀⡀⠀⠈⣆⠀⠻⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⣦⡆⠀⠀⠚⢤⠂⠀⠀⠀⠀⠀⠣⡀⠀⣀⡠⠄⠌
⠀⠀⠀⠀⠀⠈⠙⢦⣈⡇⠀⢼⠀⠀⢽⠀⡠⠿⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣰⠆⠀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⡒⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠛⠛⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣼⠏⠀⠙⠳⢶⣤⣤⣀⣀⣀⣤⣴⠶⣆⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣸⡏⢠⡶⠶⠶⣦⡀⠀⢩⡽⠷⠶⢦⡀⢹⣆⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⣿⠀⠀⠀⢸⡟⠛⢻⡇⠀⠀⢸⡟⠛⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠘⠳⠶⠲⠛⠁⠀⠈⠳⠶⠶⠟⠁⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠶⣤⡀⠀⠀⠀⣀⣤⠶⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⠙⠛⠛⠛⠉⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠟⠙⢷⡄⠀⠀⠀⠀⠀⢠⡾⠋⠻⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠶⠛⠛⠋⠉⠁⠹⣆⠀⠀⠙⢦⡀⠀⢀⡴⠋⠀⠀⣰⠏⠈⠉⠉⠛⠛⠶⢶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢹⣶⡟⠁⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠳⣄
⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣴⣿⣭⣿⣦⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡆
⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣷⠶⠛⠛⠷⢦⡀
⠙⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠟⠁⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠹
⣶⣶⣶⣶⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣦⣦⣤⣴⣶⣶⣶
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⡄⢠⡀
⠀⠀⡞⠂⠐⠂⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢢⡈
⠀⢼⡙⠁⠀⣀⣴⡇⠀⠀⠀⠀⠀⣠⠔⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠑⠀⠀⢰
⠀⠘⠉⠙⠛⢻⠋⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠑⢄⠸⡆⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠈⢣⡇⠀⠀⠀⢠⣀⠀⠀⠄⠐⠈⠀⠀⡀⠈⠄⠠⠖⠂⠁⠈⠓⠒⠄⢀⡀
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⠰⡴⣒⣷⡤⢤⣀⣀⡀⠢⠠⠂⠀⠀⠀⠀⠀⠀⠀⡤⠂⡄⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠀⣀⣿⣿⣇⠀⠀⣿⣿⣭⣁⣐⣦⠀⠀⠀⠀⠀⠀⠀⢣⡴⠃⣾⠈⢟⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⡖⠬⢉⠉⣻⢷⡾⢛⣉⠀⠤⢀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠳⡴⠃⠀⢸⢷⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣧⣀⢀⢀⠀⠀⠀⢠⢐⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠁
⠀⠀⠀⠀⡄⠀⠀⡆⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠚⠿⠖⠒⠛⠉
⠀⠀⠀⠀⠀⠀⢰⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡀⣆⠀⠀⠀⢀⡀⠀⠈⠉⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠐⢀⠄
⠀⠀⠀⠀⠀⠐⢵⣽⣶⣄⣀⠉⠁⠀⠀⠀⠀⠀⠛⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠐⠀⠀⠀⠀⡀
⠀⠄⠀⠡⠄⢤⠤⠤⠤⠄⠀⠈⠒⠀⠀⠀⠀⠀⠀⢄⡀⢠⣐⡝⠀⠀⠁⠀⠀⠀⠀⠀⠀⢴⣀⡸⠶⢦⣤⣤⣤⣀⡀⠃⢤⣤⠤⠄

⠀⠀⠘⠀⠁⠀⠁⠀⠀⠀⠀⢠⡞⣟⢷⡀⠀⠀⠀⠀⣠⠴⣶⡶⢤⡀⠀⠀⠀⢠⣶⣾⠒⢲⠀⠀⠀⠀⢀⣀⣀⠴⠦⡀
⠀⠀⠰⠴⠶⠶⠆⠀⠀⠀⠀⠈⢷⣶⣿⡇⠀⠀⠀⠀⠀⠫⠿⠯⠃⠀⠀⠀⠀⠘⡏⠙⣶⡾⠀⠀⠀⠀⠘⠛⠈⠢⢴⡃
⠀⠀⠛⠛⠀⠚⠛⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠤⢤⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀
⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⠀⣸⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⣠⢤⠀⠀⠀⠀⠀⢰⡶⠶⣶⣶⡆
⠀⠀⠻⠿⠿⠿⠟⠀⠀⠀⠀⠀⢸⣿⣽⠀⠀⠀⠀⠀⣽⣟⣿⣿⣿⡅⠀⠀⠀⠀⠸⠀⠝⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⣴⢲⡄⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣾⠛⠻⣷
⠀⠀⠁⢰⠚⡆⠀⠀⠀⠀⠀⣰⠞⢾⣻⣢⠀⠀⠀⠀⢾⣿⣿⠆⡠⠁⠀⠀⠀⠀⢸⣸⣿⡇⠀⠀⠀⠀⠀⣷⣈⣤⣿
⠀⠀⠂⠀⠛⠐⠒⠀⠀⠀⠀⠘⠲⠚⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠿⠶⠾⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣀⣀⢀⡀⡀⣀⢀⠀⠀⠠⠀⣀⢄⡀⠀⢀⠀⠀⠀⠀⢰⣶⣶⠀⠀⠀⠀⠀⠀⡟⠀⠀⡇⠀⠀⠀⠀⢠⣤⡤⠞⠛⠃
⠀⠀⠁⠉⠉⠉⠈⠀⠀⠀⠈⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣷⡤⣤⡇⠀⠀⠀⠀⠈⠋⠓⢦⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠶⠶⠶⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⢋⡩⠄⠒⠚⠋⠉⠉⠁⠒⠀⠈⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡤⠚⠁⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠈⠐⠈⠙⣦⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⡴⠋⠀⠀⠁⣀⠤⠐⠉⠁⠀⠀⠀⠀⠈⠑⠒⠄⠀⠐⠀⣲⣟⠭⠭⠭⠭⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣵⠏⡠⠀⢀⠤⠊⠁⠀⠀⠀⠐⠈⠁⠀⠀⠀⠐⠀⠀⢀⠐⣼⣋⠤⠀⠒⠀⠀⠀⠈⠺⣻⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⣿⣡⠊⢀⠔⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⠥⠒⠉⠀⠀⠀⠄⠀⢀⣿⠋⠉⠉⢩⣷⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣳⣿⠞⣡⡞⢁⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⡧⠂⢀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⢸⣿⢘⣯⡿
⠀⠀⠀⠀⠀⢀⡠⠖⠚⠛⠛⠉⠉⠉⠉⠉⠳⢆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠋⠁⠀⠀⠀⠀⠀⠀⢐⣿⣆⢀⠀⣨⣿⡿⠛
⠀⠀⢠⠖⠋⠁⣠⣴⣶⣾⣷⣶⣶⣤⣴⣦⣄⣈⠠⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡎⠀⠀⠀⠀⠀⠀⠀⢈⠀⣻⣿⣿⡯⠞⠁
⠀⠀⣟⠀⠀⢸⠁⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡐⠊⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠸⣯⡤⠀⠀⠀⣀⠠⢂⠤⣺⣿⣿⠃
⠀⠀⢿⠀⠀⠸⡄⠀⠀⠀⠀⠉⠝⠿⣿⡿⢿⣿⣿⣿⣿⣶⣄⣀⠰⠀⠀⠀⢀⡀⠤⠐⠂⠉⠠⠖⣩⣷⣖⣚⡩⠔⣊⡵⠪⣿⠟⠁
⠀⠀⠘⣆⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠉⠳⣿⣻⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⢉⠤⠒⢊⣡⠴⣻⢷⣶⣭⡤⠖⠋⠁
⠀⠀⠀⠘⢦⡀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⣻⠋⠑⣫⣷⣦⣀⡈⠡⠤⠀⠒⣉⣡⠴⣚⣥⣾⠟⠁
⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠈⠑⠒⠶⠦⠴⠷⠿⢛⣩⣴⡊⠁⣐⣨⣴⣾⠿⠻⢿⣧⣭⣭⣬⣥⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠦⠤⠶⣶⣶⣾⣯⣅⣤⣦⣵⠿⠟⠋⠁⠀⠀⠀⠀⠉⠉⠉⠉⠁

⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡃⠈⠑⠀⠰⠄⣀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠂⠢⠻
⣀⣀⣀⡄⣠⣶⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠈⠛⠲⠤⠀⠀⡙⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠴⠢⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠠⣷⡤⣤⣀⣤
⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠖⠶⠦⣤⡀⠀⢠⠆⣠⡀
⠐⠐⠀⠀⠀⠀⠀⠀⠀⠠⠨⠆⡚⠀⠢⣄⡈⢽⣷⣧⣼⠛⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠰⠏⣐⣌⣦⠘⣷⡀⢻⣿⣷⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠉⢻⣿⣤⣀⡠⠄⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡐⠺⠿⣟⣿⢃⣿⠃⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠙⣦⡀⠀⠀⣻⣿⣯⣤⡤⠖⠀⠀⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⣀⣿⣵⠟⢃⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣤⡷⣴⣾⣿⣶⣶⣿⢿⣿⣿⣶⣶⣾⠏
⠀⣧⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⢸⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠀⠀⠀⣀
⠀⣸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢰⣿⡦⢀⡀
⠀⠹⠿⢷⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⢿⠋⠀⠰⣿⢦⣄
⠀⠀⣄⡈⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢶⣶⣶⡀⠀⠀⠀⠀⣄⠀⠀⠈⡇⠀⠏⠹⢦⣄
⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⢨⣿⡀⠀⣀⣵⣴⠾⠛⠉⠁
⠠⠄⢠⣭⣥⣶⡀⠀⠀⠀⢠⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣰⣶⣾⠿⠈⠚⠉
⠀⠀⠀⠀⠘⢿⣿⣶⣶⡟⠁⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠉⠛⢿⣧⣴⡿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡐⢋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣪⠏⠈⠔⠁
⡶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠄⠀⠀⠀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢤⣤⣄⣀⣀⣼⣷⣦⣤⡐⠀⠀⠀⠀⢀⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣞⡡⠋⠁⢀⣦⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⢿⣿⣭⠤⠤⢴⡻⠿⠋
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⣇⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣽⠷⠌⠑⠄⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⣻⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⡉⠉⠻⢿⣷⣄⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⡤⠿⠆⣔⣀⣐⣒⣚⣓⡛⣓⢁⡚⡿⢽⣯⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢍⣁⣒⠂⠀⠐⡐⠀⠄⠀⠀⠠⠟⢤⣤⣤⢤⣈⣿⣿⣿⣿⣽⣴⢶⡴⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⠁⠈⢉⡉⠉⠉⠉⠉⠉⣉⠉⢉⠉⣉⣉⣉⣩⣭⣬⣶⣿⣿⣿⣽⣿⣷⣿⠺⢏⠓⣹⣒⣦⣀
⠀⠀⠀⠀⠀⠀⣴⠓⠀⠀⠡⠖⡤⠋⠸⠶⠶⠾⠧⢭⠭⠭⠭⢭⠼⣿⠷⠷⠿⠛⢻⣿⣟⣵⣿⣿⣿⣿⣶⣤⣌⡉⠉⠹⢷⣤⡀
⠀⠀⠀⠀⠀⡜⠁⣁⠠⠤⣤⣄⣰⣀⣂⣒⠒⡖⠠⢴⡆⠠⠤⠤⠤⣴⣶⣶⣶⣾⣿⠿⢾⣿⣿⢾⣿⣿⣿⣿⣿⣿⣶⣤⡈⠙⠻⣤⡀
⠀⠀⠀⠀⢰⠁⠠⠂⠋⠉⢉⣉⣙⣉⡉⠉⣉⣉⠉⡙⣛⠙⠋⠛⢉⣉⢉⣉⣤⣤⣶⣶⣾⣿⡥⢾⣿⣿⡿⣿⣿⣿⢿⠄⠁⠀⠀⢘⣿⣄
⠀⠀⠀⠀⡁⠀⠐⡒⠠⡅⠭⠉⢽⡯⠥⠬⠤⠤⠭⠭⣽⡭⠭⢬⠭⣭⠾⡷⠶⠒⢻⣟⣿⣿⠖⣺⣿⡿⠿⠿⣿⣿⡿⡶⠤⠀⠀⠨⠛⢻⣇
⠀⠀⠀⠀⡅⠐⠀⣸⣤⡤⠤⠔⠲⠶⠶⠶⠶⠶⠲⠶⠶⠦⠤⠤⠤⠴⠶⣶⣶⣿⣿⣿⡿⠖⣊⣿⣿⣧⡤⠶⣼⣽⣭⣓⣀⠀⠀⠀⡀⠀⢻⣄
⠀⠀⠀⠀⡀⠒⡫⢽⡶⠒⠒⠛⠉⢻⡋⠉⠛⠛⠛⠛⠛⣟⠉⠉⠙⣋⣉⣋⣩⣬⣼⣶⣟⣋⣿⣿⣿⣿⣿⣷⣿⣷⣾⠦⢈⣉⠀⠀⠈⡄⠠⡻⡄
⠀⠀⠀⠀⠄⠀⡀⢘⣫⡦⠒⠒⠐⠀⢒⠀⠀⠀⠀⠉⠉⡨⢁⠀⣁⣉⡩⠴⢶⠾⣛⣋⣭⣿⣿⣿⣿⣻⣿⣿⣿⣿⣶⠖⡢⠄⠈⠑⠢⠤⣀⡱⠁
⠀⠀⠀⠀⢖⢰⡎⢀⡘⠧⣀⠠⠤⠐⠀⠆⠀⠀⢀⣀⣈⣹⡤⠤⠤⠴⠶⣶⡟⢍⣱⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣡⠠⣄⡀⠀⠀⠰⢌⣁⣃
⠀⠀⠀⠀⢠⡀⡡⠘⠉⠡⣤⠤⠀⠒⢂⣉⣁⣠⠤⠤⠤⠶⣶⣐⣖⣻⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣟⠛⠛⠿⢶⣬⡁⠀⡀⡀⠀⠋⠆
⠀⠀⠀⠀⠀⢌⠀⡸⠒⢀⣨⠶⠒⠉⣉⢀⣤⡤⠶⠶⣒⣛⣻⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣁⣉⠐⠂⢄⡙⠳⣦⣉⡒⠐⡄
⠀⠀⠀⠀⠀⠀⢃⣴⠚⢉⣠⠶⠞⢂⣩⣥⠴⠶⣊⣩⣥⣚⣾⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣻⣻⣿⠿⣿⣿⣿⣛⠋⠙⠛⠻⠶⣤⣀⠓⠤⣙⠛⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠣⣴⠟⠁⣠⠞⢋⣡⠴⣚⣭⣴⣾⣿⣿⠿⠛⢋⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠅⢀⠀⠦⣄⡀⠈⠛⠿⣦⣌⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⡞⢀⡴⢋⡵⢛⡩⣛⠿⠛⠉⠅⠀⣤⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠂⠀⠀⠀⠲⣤⣍⠳⣦⣀⠈⠙⢷⣦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⣱⡟⣠⢟⡕⢡⠆⠀⠂⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⡶⢤⣀⠀⠀⠈⠻⢿⣦⣍⠻⢦⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢏⠺⡐⠀⠈⠠⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠈⠻⢿⣶⣤⡐⠠⡈⠛⠿⣦⡉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⡙⢿⣿⣿⢿⣿⣿⣿⣿⣟⣛⢿⠿⣿⣿⣍⠀⠈⠻⠓⣦⣀⠀⠉⠙⢿⣷⣄⡀⠀⠈⠻⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠈⢰⣷⡀⠹⣿⢸⣿⣿⣿⣿⣿⣻⣿⠷⠶⠀⠀⠈⠐⠀⠀⠀⠈⠛⣦⣀⠀⠉⠛⢿⣶⣄⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡄⢀⠸⣿⣷⡄⡟⣼⣿⣿⣿⣿⣿⣿⣿⠷⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠓⢦⣀⠀⠘⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡈⠀⣿⣿⣧⢀⣿⣿⣿⣿⣿⣿⣿⡷⢄⡀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠁⠀⣀⠈⠝⣳⡄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡧⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⡻⠟⠣⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢤⡐⠈⡿⣦⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣿⣿⣿⣿⣿⣿⣿⣿⣯⡛⢦⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢤⡀⠈⠙⢦⣄⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠘⢿⣿⣿⣿⢿⡝⢿⡛⣆⠈⢢⡀⠈⠳⣄⠀⠑⣆⠀⠀⠻⣦⡀⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠀⣿⣿⣿⣧⡹⣦⢳⡌⢣⠀⠱⡄⠀⠹⣦⠀⠸⢧⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠇⠀⠉⠻⣧⠘⡇⢻⡄⢳⡀⢹⡆⠀⢹⣧⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠈⠃⣹⡈⡇⠈⣷⡀⣿⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣴⠖⠞⠋⠀⢸⣿⣦⡶⠒⠂⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⢔⡫⠉⠙⢿⣿⣿⡆⠆⠀⠀⢸⣿⣿⠁⠀⠰⣿⣿⣤⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠸⣿⢷⣿⡆⡄⠀⢸⣿⣿⣧⠀⠀⠀⠨⣯⣤⠂⢀⠈⣿⣿⣿⠦⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⡞⠃⠀⠆⠈⠈⣿⣧⢀⠀⠘⣻⣿⡇⠀⠀⠀⢰⣿⡿⠂⠀⠀⢸⣿⡝⠀⢈⣻⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠙⢿⣿⣿⡇⠀⠀⠀⣦⢹⣤⠘⠀⠀⣼⣿⡇⣾⠀⠀⠌⣿⣟⠀⠀⠀⢻⣿⠃⠀⠠⡿⣿⣿⠏⢰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠤⠁⠹⣯⡀⠀⠀⠀⠘⣿⡇⠀⠀⢹⣿⣗⣿⠀⠀⢀⣿⠘⠀⠀⠀⣿⡟⠀⠀⢦⣟⣿⡏⠀⢨⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣷⣿⢦⡀⠀⠀⠀⠘⣿⣀⡀⠀⣦⢹⣧⠀⠀⠘⣻⣿⣯⠀⠀⢨⣯⠀⠀⠀⢰⣿⠃⠀⠀⣸⣿⡟⠀⢀⣿⣾⢿⢟⣍⡂
⠀⠀⠀⠀⠀⠀⢠⡝⠛⠛⢾⣷⡄⠀⠀⠀⠙⣿⣇⠀⠀⠈⢿⣆⠀⠀⣿⣿⣿⠀⠀⢸⣿⠃⠀⠀⣾⡏⠀⠀⠀⢿⡿⠁⠀⣿⣿⡿⠃⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠰⣧⠀⠀⠀⠻⣿⣦⠀⠀⠀⠹⣿⡆⠀⠀⠸⣿⠇⠀⢐⣿⣿⠀⠀⢸⣿⡁⠀⢀⣿⠇⠀⠀⣴⡿⠃⠀⢠⣿⡿⠁⠐⠂⣤⣿⡀
⠀⠀⠀⠀⠀⢀⣿⣿⣧⡄⠀⠀⠈⢿⣷⠀⠀⠀⠹⣿⡄⠀⠀⢿⣧⠀⠈⣿⣿⠀⠀⣠⣯⠁⠀⢸⣿⠀⠀⢠⣿⠃⠀⢀⣿⡟⠀⠀⢘⣾⣏⣽⡟
⠀⠀⠀⠀⠀⠈⠟⠛⠛⣿⣷⡀⠀⠀⢹⣷⡄⠀⠀⠹⣿⣄⣠⣼⣿⡆⢸⣿⣿⡇⢘⣯⣿⡀⢰⣿⡇⠀⠀⣾⡏⠀⢰⣿⠋⠀⠀⣼⣾⡿⠙⣱⣷
⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠙⢿⣦⠀⠀⠈⢿⣄⡀⣶⣿⣿⡟⠉⢻⣇⠈⢹⣿⠇⠀⢹⡿⠀⢸⣿⠉⠘⣾⣿⢣⣶⣿⡏⠀⢀⣰⡾⠋⠁⣹⡿⣻⡀
⠀⠀⠀⠀⠀⠀⡏⠣⣿⣤⡀⠀⠙⢷⣆⠀⠘⢿⣧⡀⠙⠹⣷⠀⠘⣿⠀⠀⣿⡃⠀⣸⡇⠀⢸⡏⠀⢰⣿⠁⢐⣿⠋⢠⣷⣿⡟⠁⢠⣾⠟⠐⣻⠇
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠙⠷⣄⠀⠀⢿⣿⣆⠀⠈⠳⣄⠀⠹⡆⠀⢻⡇⠀⣿⡇⠀⣿⠀⠀⣾⠁⠀⣾⠃⢀⣾⠃⠀⣸⣟⢻⣦⣴⠟⠁⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⣼⣿⣿⡀⠀⠙⢦⡀⠀⠙⣆⠀⠹⡄⠈⣷⡄⢺⣷⠀⣿⠃⢀⡟⠀⣾⠇⢀⣾⠃⢀⣾⡿⠁⣴⡿⢿⣧⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⡈⠉⠋⠙⢦⡀⠀⠙⢦⡀⠈⢳⡀⠹⣆⢸⣇⢸⣿⠀⣿⠀⣸⠃⣼⡏⢀⡾⠃⢠⣿⠋⣀⡾⠋⡀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠙⠲⣄⠀⠙⢦⡀⠹⣄⠻⣆⣿⡌⣧⡄⣿⠀⣿⣰⡿⣠⣾⠃⣰⡟⢡⣼⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠈⠳⢤⡀⠙⢦⡘⢣⣿⣿⣷⣿⡇⣿⢸⣿⣿⣷⡿⢃⣴⢏⣴⢟⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢆⡀⠀⠙⢦⣤⡙⢦⡙⣧⢻⣿⣷⣿⣼⣻⣿⡞⣡⠟⣡⢿⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣤⡀⠙⠿⣾⣿⣮⣷⣿⣿⣿⣿⣟⣾⣿⣿⣾⠗⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣉⠛⠶⣦⣌⡛⢷⣿⣯⣿⡿⣿⣿⣟⣿⠟⠁⠀⠈⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣮⣉⠉⢁⠂⠹⢿⣧⣝⣿⣿⣿⣿⣿⡿⢃⠁⠀⠀⠘⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠅⠤⠤⠀⠀⠚⠛⠛⠛⠛⠛⠟⠋⠉⠉⠁⠐⠒⠒⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣟⣛⣿⣛⣛⣿⣭⢯⣽⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣀⣀⣠⡤⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⢿⠿⠛⠁⠠⣤⣤⣤⣤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡆⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣇⣀⣀⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣿⠿⢿⣿⣿⠀⠀⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠋⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⠇⠀⠀⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⣿⣤⣄⠀⢿⡿⠀⠀⠀⢸⡿



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⣤⡤⣀⡀
⠀⠀⠀⠀⡈⠘⢄⠠⢔⡲⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡞⠓⢺⣅⠀⠏⣷⣰⠉⢲
⠘⠯⠭⣉⠁⡷⡆⣰⠋⠀⠀⠀⣰⢹⣭⡭⠷⠦⢙⣶⣶⣶⠆⠀⠀⠀⣀⣀⠠⢤⣀⠀⠘⢯⡿⣿⠇⢰⢻⡁⠠⣁⣿⣿⣆⠘⡇
⠀⠀⠀⡜⣠⠖⠢⡘⣆⠀⠀⠈⠻⣇⣛⣛⣋⠉⣈⠿⠋⠀⠀⢀⠴⠿⠤⣍⢢⡀⣿⠈⢢⡀⠉⢿⡇⠘⣗⠀⢘⢠⡅⠙⢿⣶⣿⣄
⠀⠀⠈⠚⠁⠀⠀⠙⠚⠃⠀⠀⠀⠘⠛⠒⠂⠉⠀⠀⠀⠀⠀⢮⣿⣿⣷⣮⣿⣧⣇⡠⠏⠁⠀⠈⠋⠙⣯⠀⣰⣰⣀⡴⠶⠻⠿⣿⡧
⠀⠀⠀⣀⢤⠴⠶⡶⡤⣤⡀⠀⠀⠀⠀⠀⠀⢀⠠⡄⠤⢤⠠⠀⠀⠉⠙⢫⠞⠻⣟⠒⠒⡏⠀⠀⠀⠀⠈⠋⠉⠉⠁⢠⡴⠶⢤⡀
⠀⣴⠻⠹⠀⠀⡄⠀⢀⠃⠞⣳⣄⠀⠀⡠⠊⠣⡀⢣⡀⣹⠀⠇⠀⠀⠀⠘⠣⣧⠈⠹⠞⠀⠀⠀⠀⢀⡄⠀⠀⠀⡠⣿⢏⠢⠡⢹⡄
⠈⡎⢂⠁⠃⠀⢀⠀⠈⠈⡰⠡⢋⠆⡐⠙⠢⣤⠟⠉⡏⢻⢆⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⣸⣷⠀⠀⠀⡇⡈⠘⡄⡂⠆⣧
⠀⠘⢆⢢⢠⣠⢘⠀⠀⠐⠐⡡⠋⠀⡗⠲⠴⡟⢒⣶⡷⣯⠤⡇⠀⠀⣰⠇⡢⠀⠄⡁⠨⢳⡄⠀⠀⡯⠰⡄⠀⠀⢻⢁⠀⠀⠀⠀⢻
⠀⠀⠀⠳⣕⢷⣿⣾⠌⣠⠊⠀⠀⠀⠘⢦⠒⢛⠥⣮⢶⠛⣤⠇⠀⠀⣥⠊⣠⠎⡔⢠⠁⡅⡇⠀⣘⣋⠉⢵⠀⠀⠸⡼⡠⡀⠀⠀⡿
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠈⠙⠻⠤⠧⠜⠛⠁⠀⠀⠀⠋⣴⠃⣼⠁⣾⠀⣇⡿⠀⠫⣤⣺⣿⡷⠀⠀⢿⡵⣣⢇⢢⡇
⠀⠀⠀⠀⣠⡤⣀⢤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠒⠛⠴⠟⠒⠉⠁⠀⠀⠸⠿⠋⠁⠀⠀⠈⠻⣵⣯⠞
⠀⠀⡄⠊⠀⢳⡀⠈⢁⡆⠙⢦⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠻⣟⠛⠒⠚⠳⠲⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⣼⣀⣀⣠⡜⢋⠓⠺⢀⠠⠈⢳⣄⠀⠀⡼⣛⣫⡈⠛⣷⣄⠀⠀⠉⢢⡀⢀⠀⠃⠈⠙⠦⣄⠀⠀⠀⠀⠀⢴⣾⣿⣿⢧⣄
⠀⢱⠀⠏⣇⡼⣟⡖⠈⢹⠶⠛⠉⣿⠀⠀⡏⣴⣾⣿⣮⡟⣿⣆⠀⠀⠀⠙⠒⠒⠓⠒⠒⠒⠚⠓⠀⠀⠀⣠⡾⣿⣛⣫⣤⠛⠙⡦⡀
⠀⠘⣦⠚⠙⠑⡎⠉⠺⡇⢠⠀⣆⢺⠀⠀⢣⡻⣿⣿⣿⣿⡿⠛⠀⣀⠤⢤⠞⠛⠙⣟⠓⠦⣄⣀⠀⢠⢞⡙⠀⠈⠀⠀⠀⠐⡜⡄⢽
⠀⠀⠈⠙⠲⠶⠧⠖⠲⠶⠚⠒⠛⠛⠁⠀⠀⠙⠚⠛⠋⠉⠀⠀⢰⡡⠊⠀⢂⠀⠀⠘⣄⠳⣈⡟⠀⠸⡕⢡⢀⠁⠃⡛⡈⡁⡐⡜⣼
⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣔⣚⠉⠓⢔⠄⡄⠈⣳⠋⠀⠀⠀⠘⠧⣼⢐⡃⡃⡇⢡⣱⡴⠋
⠀⠀⠀⣴⣿⣿⣛⡷⡄⠀⠀⢀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⠈⠳⣝⣛⢛⣿⣚⣷⠤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠛⠉⠁
⠀⢠⢚⢝⡉⠀⢉⣶⡿⡆⠀⡟⡩⠋⡥⠼⣧⡈⠉⢀⣴⣦⡌⢷⡀⠈⠈⠉⠀⠉⠀⠀⠀⠰⡒⡖⡲⠉⡟⠉⢹⠖⠘⠳⢦⣄
⠀⣇⢏⢮⠸⣼⣿⡿⣿⠁⢠⡋⢰⠺⢷⡄⠈⠻⣦⠈⢿⣿⣿⡎⣧⠀⠀⠀⢀⡖⠒⠦⣄⠀⠘⢶⠃⢀⠀⠀⠀⠀⢐⠈⠀⢘⢆
⠀⠈⠲⢽⣽⣿⣫⡞⠁⠀⢈⣶⡛⠳⣆⠙⠷⣄⠙⣦⢸⣿⣿⣧⣽⠀⠀⣀⡼⠅⠀⠀⠠⠛⣄⠈⠛⣇⠀⢠⠀⠀⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠘⢏⠙⠶⢌⣙⡓⢮⣿⣌⢣⣻⣿⣯⠏⠀⢸⠙⠌⠀⠀⠀⠀⠄⢹⠀⠀⠈⠛⣇⠀⠐⠀⠀⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣭⣭⣴⣍⣰⣇⠉⣿⡟⠀⠀⠈⠣⣀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠘⢶⣄⡀⠀⠀⠠⠀⢛⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠚⠓⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠦⠉⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣞⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡄⠘⠋⠁⢀⣌⣻⣷⡄⠀⠀⠀⠀⠀⢀⣤⣤⣶⣦⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣏⣠⣾⠟⢲⣯⠀⢡⣈⢷⣄⣀⣀⣴⣟⠍⠑⠒⢞⠁⢠⣉⠩⡉⠛⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢿⡇⠈⠉⠅⠀⠀⢉⣶⢿⣿⠛⠛⠈⠻⣧⡫⢀⣀⣀⠀⠀⠀⠁⠠⠉⢂⣶⠙⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣿⣷⠆⠐⠀⠀⠀⣠⣿⣿⣄⡉⠶⠀⠀⢀⣈⣙⣻⠿⣽⣇⠀⡀⠀⠀⠀⠘⠇⠐⢻⡿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣥⣄⡀⠀⠈⣿⡷⢟⠉⠙⠿⠶⠤⠤⠈⠉⠉⠁⣰⠛⠙⣿⣿⡐⠀⠄⠀⠀⠀⢨⡼⣯⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠛⠿⢷⣶⣀⠁⢀⠼⠋⢀⣬⡧⠄⢠⣄⣀⠀⠀⠀⠀⢀⡽⠀⠀⣠⣿⠣⣄⠀⠀⠀⠀⠈⠀⣿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠺⣏⣹⣿⣿⡶⣄⠀⠀⠀⠒⠀⠀⢀⠋⠉⠀⠀⠀⠀⠄⠀⠀⠀⢰⢲⠗⢠⣾⣿⣷⠟⣿⠀⠀⠀⠀⠀⠀⠟⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣧⠹⠁⠀⠀⠀⠀⠀⢀⡀⠈⠆⡠⣿⠐⠀⠀⠀⢀⠙⡧⠀⣾⣿⣿⣧⣼⠟⠀⠀⠀⠀⠀⠀⠸⠘⣷⡀
⠀⠀⣠⡶⠾⠛⠋⠁⢐⠿⡿⣿⣇⠀⡀⠀⠀⠀⠀⣴⣶⠞⠃⠀⠀⠉⠁⠀⠀⠀⡀⠈⠁⣸⣿⣿⣿⣿⡅⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⢿⣿⣷⣤⣤⣐⡫⣄⣴⡿⣿⠀⡄⠀⠂⣀⣀⣸⢽⠃⢐⠓⠘⡆⠀⠀⠀⠃⠁⠀⠀⣿⣏⣿⣿⡏⠹⠂⠐⢀⠀⡀⠀⠀⠀⠀⠀⣹⡇
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⢴⣿⣀⣠⣶⠂⠀⠈⠁⠀⠀⠈⠀⠀⡇⠀⠀⠀⠀⠀⠰⠀⣿⣿⣿⣿⣇⠲⡄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄⢻⡄
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⠶⢀⣀⠀⠠⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⡯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣻⣷⣿⡿⠇⢀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣟⣋⠥⢿⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⣿⡟⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣈⠠⣀⠻⠂⠀⠀⠀⠀⠀⠀⠀⣼⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⡈⢁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣉⣉⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⣛⣿⣜⠓⢉⠠⠀⠈⠀⠀⠀⠀⠀⢰⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⣻⣾⣧⡄⠀⠀⠀⠀⠀⠀⠀⣼⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⠀⠈⠄⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣟⣯⠙⣿⣄⠀⠡⠀⠀⢠⡿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⠧⣤⣀⠰⠄⠀⡀⡇⣸⡗⠈⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡿⠶⣤⣏⣉⠶⢢⡾⢁⣿⠟⠃⠀⠀⠀⠀⠀⡀⣀⣨⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠿⠿⠛⢨⡤⠛⢀⣿⡿⠟⠀⠀⠀⠀⠀⠀⠙⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠃⠔⠀⢠⡄⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠠⢴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡎⠐⣀⣶⢋⣐⣿⠃⠀⠀⠀⠠⣤⣤⣀⣤⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣁⣺⠟⣡⣾⠿⠁⠀⠀⠀⠀⠀⠹⡄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣙⣿⡆⢿⡿⡃⣀⣀⣀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⢾⣷⣌⡿⠛⠉⠉⠙⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠛⠃



⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⡿⠿⠿⢿⣿⠿⣿⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⡤⢤⣠⡤⠶⠤⠤⠤⣤⣀⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠋⠉⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡏⠋⠉⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣾⡟⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣷⣲⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⢸⣿⠇⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢙⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⠋⠀⠀⠈⠉⣧
⠀⠀⠀⠀⠀⠀⢀⣀⣴⡏⠀⠀⢸⠻⠀⠀⠀⠀⠀⠀⡏⠀⠶⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢀⣯⣿⡳⢦⡀
⠀⠀⠀⠀⠀⣴⠋⠙⣿⡄⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢣⠆⠀⠀⠀⢸⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⣸⠈⠤⠀⠀⢀⡾⠉⠁⠀⠈⢻⡄
⠀⠀⠀⠀⢰⠗⠀⠀⠹⡇⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢸⠠⠀⠀⠀⢸⡄⠀⠀⠀⠀⣿⠀⠀⠀⢀⣧⠀⠀⠀⣠⣟⡈⠀⠀⠀⠀⣼⣅
⠀⠀⠀⣠⡿⡤⠀⠀⠀⢣⡀⠀⠀⠆⠸⠀⠀⠀⠀⠀⠸⡟⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⡸⡏⠀⠀⢈⡿⠉⠀⠀⠀⠀⣼⡿⠛⢧
⠀⠀⠀⡿⠀⠘⡆⠀⠀⠀⢡⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡆⠀⠀⠀⢰⠃⠀⠀⢠⣇⠀⠀⠀⡼⠀⠀⠀⠀⠀⡜⠁⠀⠀⣸⡁
⠀⠀⢠⡧⠀⠀⠈⢷⣄⠀⢈⢧⡀⠀⠀⠘⡄⠀⠀⠀⡀⢠⣆⠀⠀⠀⣷⠀⠀⡀⣸⡄⠀⠀⡜⠀⠀⠀⢰⠁⠀⠀⠀⠀⡜⠀⠀⠀⣰⢿⢿⡀
⠀⠀⢠⡿⢄⠐⠀⠫⣻⣄⠀⠈⢧⡀⠀⠀⢹⣼⠀⠀⠀⢸⠸⠀⠀⠀⡿⠀⠀⢠⡿⠇⠀⢰⠛⠁⠀⣠⠧⠀⠀⠀⢀⡜⠀⠀⢀⠜⠁⠀⣠⠇
⠀⠀⠘⣧⠀⠳⡠⠀⠙⢿⡆⠀⠀⢻⡧⠀⠈⢿⠇⠀⠀⠐⡎⠃⠀⠀⠀⠀⠀⢸⡏⠀⢀⣿⠀⠀⢀⡏⠆⠀⠂⢀⣾⠀⠀⢠⠎⠀⢀⡶⢹⡆
⠀⠀⠀⢿⠻⣶⡿⣦⠀⠀⠙⢶⣄⠀⣿⡀⠀⠘⡄⠀⠀⠀⢣⠀⠀⠀⡀⠀⠀⣸⠀⠀⣸⠏⠀⢀⣾⠂⠀⠠⣠⡃⠄⢀⡔⠁⢀⣞⡩⠒⡏
⠀⠀⠀⠈⢳⣌⢻⣌⠳⡄⠀⠈⢿⡄⠀⢳⡀⠀⢹⡀⠀⠀⠸⠀⠀⠀⠁⠀⠀⡟⠀⢀⡇⠀⠀⣾⠃⠀⢠⡞⠁⠀⣠⡊⢊⢔⣵⢋⡥⣴⠃
⠀⠀⠀⠀⠘⢮⡻⣍⠢⡝⢦⣀⠈⢻⡆⠀⠹⣄⠀⢣⠀⠀⠀⡇⣄⠀⢠⡤⣠⢇⠀⣼⠀⠀⡼⡕⠀⢀⡏⠀⢀⡾⢫⢞⡵⣫⢟⣫⡵⠋
⠀⠀⠀⠀⠀⠀⠙⡮⢷⣍⡲⡙⢄⡀⠙⣦⡀⠹⡆⣸⡞⠀⠂⢻⠋⢸⠈⠁⣸⠉⢱⠿⠦⣴⡷⠀⣠⠋⢀⡴⣡⢞⡵⢛⣫⡶⣩⠟
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣍⣻⣿⣦⡙⢄⡈⢿⣦⠙⡅⢹⡀⠀⢸⡀⢸⠈⠀⡎⠀⣞⠁⢠⡿⠛⣳⣅⣠⡫⢚⢕⠫⣖⣫⠵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣭⣝⡳⢷⣦⡙⢧⡘⣄⢣⠀⠈⡇⢸⡄⢰⠁⢰⡏⣠⡏⢁⡼⣩⣻⣿⣾⡿⠗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣛⢷⣯⣿⣅⠹⣜⢆⢇⠀⣧⢸⡇⡼⠐⣾⢧⡿⢡⣾⣿⣿⣽⠟⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣈⠉⣽⡍⠉⠛⣷⣽⣮⢻⣧⢸⠸⢷⣧⢲⢏⡿⣱⣿⡿⠿⠟⣫⠴⠞⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡘⠛⢳⣴⣶⣄⣸⡻⠟⢷⣿⣄⡆⣼⢇⣯⢎⣾⡿⢿⠁⣀⢶⣿⣧⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⢉⣒⠀⠀⠀⣨⡙⠓⢚⣿⣿⣧⣿⣾⣿⣿⣟⣂⣿⣏⠩⣭⡯⠶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠙⠛⠛⠉⠉⠉⠀⠈⠙⠛⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠛⠁
⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡯⡄⢠⠀⠙⢿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⢰⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠸⡿⠷⠆⠀⠻⠿⠟⠁
⠀⠀⠀⠀⠀⣿⣿⣟⣛⣛⣛⣛⡃⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⢂⣀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠾⠷⠆⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣭⣭⡅
⠀⠀⠀⠀⠀⣿⣿⡿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣶⣦⠀⠀⠀⠀⠠⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠴⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⣿⣿⡇⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⣿⣿⣯⣭⣭⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣧⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠻⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡞⠉⢹⡗⢻⠀⠀⡟⢻⠁⠉⡱⢰⠆⠀⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡤⣏⠀⢻⠀⢷⠀⠈⡇⢸⠀⠀⡇⢸⠀⢀⡇⣸⠀⠀⡏⢹⠁⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢶⢻⠀⢱⠸⡀⠈⣇⠸⡄⠀⡇⢸⠀⢠⠇⣿⠀⢸⠃⡏⠀⢰⢁⠇⠀⡼⣹⠋⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⣄⠈⢇⢣⠀⣇⢇⠀⠘⡄⢧⠀⢿⠸⡀⢸⠀⣿⠀⣼⢰⠃⠀⡏⢸⠀⣰⢡⠇⢀⡞⢳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠹⣆⠘⣆⠘⣎⣧⠸⡘⡄⠀⢳⠘⡆⢸⠀⡇⢸⠀⡇⠀⡇⣼⠀⢰⢃⡇⢠⢃⡞⠀⡼⢠⠃⡸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡙⣆⠘⢦⠈⢦⠘⣞⡇⠡⠹⡀⠸⡆⢸⠈⡇⣇⢸⢀⡇⢰⠁⡇⠀⡟⢸⢀⡎⡸⠀⡸⢡⠇⡰⢡⠏⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠙⣞⢧⡈⠳⡈⢧⠘⡾⡄⢇⢣⠀⢣⠘⡆⡇⢹⢸⢸⠀⠸⢰⠂⢸⢁⠇⡼⢠⠁⣰⢡⠏⡼⢡⠃⢀⢞⡷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠟⢷⡄⠈⢧⡱⣄⠙⣄⢣⠱⡹⡜⡆⢇⠸⡄⡇⠁⠈⢸⢸⠀⡇⡸⢀⡏⡜⢰⢁⠎⢰⢃⡎⡼⢡⠃⡰⢣⠎⢀⠿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠷⡄⠙⢦⠀⠳⡌⢦⡈⢆⢣⠱⠱⠘⡼⡀⠇⠠⢠⠀⢸⢸⢰⠀⠇⢸⢠⢃⠃⡜⢠⠇⡎⡸⣱⢃⡜⡡⠃⣠⠋⣠⣇
⠀⠀⠀⠀⠀⠀⠀⣸⠳⣜⢦⡀⠳⡀⠘⢆⠳⡈⢆⢣⢡⣃⠰⠁⠸⠸⠸⠀⠀⢸⠀⢰⠀⡆⡜⡌⡰⠁⠎⡸⣰⢱⢁⠞⡰⢁⡴⢡⠞⣡⡞
⠀⠀⠀⠀⠀⠀⠀⢹⣄⠈⠳⣜⠦⡘⢦⡈⢣⡙⡄⢻⣇⢻⢆⢂⢀⠀⡆⡀⡄⡀⠀⡘⡸⠰⠰⡱⠁⡜⠠⠡⠱⣡⢊⡜⣡⢎⠔⢡⠞⣡⡾
⠀⠀⠀⠀⠀⠀⠀⠘⡌⠳⣄⠈⠓⢌⠢⡙⢆⠱⡜⢆⢻⡆⠚⡀⢎⡄⢷⡇⡇⡇⠀⢃⢃⢧⢁⠃⡰⠡⠡⠀⡵⢡⢞⡴⡵⢋⡔⢁⠔⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠑⢦⡀⠑⢝⣆⠳⡘⢎⢂⢣⢧⢡⠸⣇⢸⣧⣇⠇⡜⡘⠈⡆⠌⡰⠡⠁⠡⠌⡰⡣⢪⢎⡵⢋⡴⢃⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠙⠦⡀⠻⣕⢝⣎⢪⢦⢻⣿⡆⣿⡄⣿⣿⢠⣳⢁⢡⡞⡰⠡⢡⢃⢎⠌⣔⢔⠕⣩⠔⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢌⠢⣈⠢⣹⣧⡻⣆⣯⣿⣸⡇⣿⣿⢸⣿⡘⣼⠰⢁⡳⢁⣤⣧⣶⠗⡡⣚⠵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢪⡑⢌⠻⣷⡹⣦⣿⣧⣿⣿⣿⣿⡿⢰⡷⢡⡾⣱⣿⣿⣿⡵⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠑⠷⣮⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣣⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠐⢀⠠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠀⠀⠤⠀⠙⢿⣿⣿⣿⣿⣿⣟⣉⣀⣀⣈⣁⣀⡶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉

⣦⠀⣴⢠⣶⣶⣶⣶⣶⣶⣶⣦⢰⡀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢿⣿⣿⡿⢶⣦⣀
⣿⣿⣿⣞⣿⢿⢿⠿⠿⡿⠿⣟⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢟⣽⣾⣿⣿⠿⢿⡷⣯⡻⣧⡀
⡏⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⣾⡩⠥⠰⠆⣻⣯⣭⣾⣷⣹⣷
⣿⣤⣿⢯⣽⣭⣭⣭⣭⣭⣭⣭⢻⣦⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⣿⡗⣛⣐⣋⣚⣃⣛⢸⣿⣏⣿
⣿⠛⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣸⡟⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⢿⡿⣿⣿⣿⣿⣿⣿⢸⡿⣹⡟
⣿⣀⣿⢸⣿⣿⣿⣿⣿⣿⡿⣻⣼⣶⣶⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣮⡻⢶⣶⣶⣶⣶⡶⢟⣵⡿⠁
⣿⣿⣿⢸⣿⣿⣿⣿⣿⢯⣾⣿⣿⣿⣿⡿⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣾⣯⣽⣷⣾⣿⡿⠁
⣿⠀⣿⣮⣿⣟⣿⣻⡏⣾⣿⠿⠶⢶⣾⣿⣷⣝⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟
⣿⣶⣿⣿⠿⠿⠿⠿⡯⢿⣿⣿⣛⣛⣻⡟⣻⣵⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠏
⣿⠛⣿⢱⣿⣿⣿⣿⣿⣮⢿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁
⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠛⠛⠛⠛⠋⠁





⢀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀
⢸⣧⣴⣿⢻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣦⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢠⡄
⠘⡟⠛⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⣛⣛⣛⡛⠿⢿⣯⣭⣭⣽⣿⣿⣿
⢸⣇⣀⣿⢟⣛⣛⣛⣛⣛⣛⣛⡻⣿⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣿⣧⣿⡧⣛⣻⣿⣿⣿⣿⣿⣿⠃⢀⠬
⢸⣿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⠠⡻⠃
⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣛⣤⣼⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡶⢶⠶⠶⠶⠶⢶⠶⠶⢾⡿⡱⡟
⢸⣷⣾⣿⣿⣿⣿⣿⣿⣿⢟⣵⡿⣿⣿⣿⣿⢿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⢟⠾⣎
⢸⡏⠙⣿⡹⠿⠿⡿⠿⠏⣿⣯⡒⣼⣿⣿⡿⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⡌⣱⣿
⢰⣧⣤⣿⣿⣿⣿⣿⣿⡟⣿⣿⡇⣿⢗⡻⣦⢡⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣟⣛⣛⣛⣛⣛⣻⣿⣤⣾⣿⣿
⢸⡿⢿⣿⣩⣿⣾⣿⣿⣦⡻⣿⣇⣿⣮⣾⣿⣸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡘⢇⠀⠟⠻⠟⠛⠛⠛⠛⠿⢎⠻⣿⣿⣿⣿⠟⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢈⠉⠛⠛⠛⢀⠀⢀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢴⠀⠀⠀⠀⢀⠁⢄⡄⠤⠂⠜⠟⣧⡆⠀⠀⡤⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦⠀⣿⣄⠀⠀⠸⣿⠂⠀⠊⠘⠁⢠⣾⣿⣷⡄⠀⣟⠀⣠⣠⠀⠀⠀⠐⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠅⡈⠀⣏⡴⣶⢀⠉⢈⣟⣶⢸⣷⠟⢈⡟⡏⢡⣆⢿⣿⣿⣿⡯⣀⠀⠀⢀⣼⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⢀⣩⠎⠀⠀⡶⣴⡿⠟⡛⠻⠀⠓⢀⣃⣺⣤⡟⣿⣿⣏⣷⣿⠀⠀⠉⠁⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⡌⠀⠀⠀⢸⣶⣾⣿⠗⣾⣷⡷⢦⣠⣯⣼⠷⠛⣿⠉⢹⣿⠃⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡶⠚⣴⣤⠀⠀⠈⣧⣼⣷⠶⣿⣿⣤⣼⠼⣿⣷⡀⠘⡟⠀⢸⣿⡁⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠋⠀⠈⠷⠐⣉⣹⣤⢤⣟⣋⡙⠳⢼⣏⣉⢿⠿⠏⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠁⠀⠠⠂⢀⠀⣍⠉⠉⠀⢉⠇⠀⠺⣿⣿⠉⡄⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠢⠀⠆⢐⠂⠀⠄⣀⢠⡀⠈⣮⣆⣣⠀⣰⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⢸⣆⣠⣄⣀⡀⠠⣀⢷⣺⣷⡤⣤⣼⣉⣷⣦⣬⣿⠀⣰⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡏⠀⠻⢍⣤⣽⠾⣄⣿⠿⡛⢋⣿⣽⣿⣾⠃⣰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠉⠸⠀⠀⣿⠻⠎⢽⡟⢢⡽⠋⠁⣴⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣧⣀⣠⡇⢤⠀⢆⢰⡇⢀⣿⣤⠟⠄⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡎⠋⢿⣣⣸⡁⢧⢇⢠⣾⣿⡏⠀⠀⣴⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡉⣼⢿⣯⣷⢲⣾⡙⠿⠏⠑⠀⢈⣿⣿⣿⡿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⣿⣿⡟⠋⠉⠀⣀⠀⣰⣿⣿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠉⠀⠀⣀⠆⠁⠐⣹⣿⣿⡟⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⠀⠀⠀⠼⠉⠀⢀⣼⣿⡿⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠈⢸⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠘⣸⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⡀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⠀⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣇⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣦⣷⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠛⠁⠛⠀⠄⠙⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠣⡀⠀⢻⡀⢠⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⣀⣀⣀⡀⠀⡄⠀⡄⠀⠀⠘⠂⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⢱⣦⡀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⣾⣽⣿⣿⣿⡎⠀⢐⡃⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠘⠳⣦⠀⠀⠀⠀⡜
⠀⠀⠀⢿⣿⣿⣿⣿⠁⠀⢸⡇⠀⠀⣀⡆⠀⠀⠀⠀⠀⢀⠀⢠⡆⠂⠀⠀⠓⢀⡈⠂⠀⠦⠀⢰⠃
⠀⠀⠀⠈⢿⣿⣿⣿⠄⠀⠀⠧⡞⠉⠉⠁⠀⠀⠀⠀⠀⠘⠊⠛⣷⣆⠈⠂⠀⠀⠉⠠⠄⠀⢠⠇
⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠐⡀⠀⠀⠀⠁⣀⡀⠠⣤⠄⠀⠟⢻⣾⠃⠀⢠⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⢹⣿⡿⢀⣀⡤⠄⣐⣓⣤⣦⣭⣭⣥⣴⠄⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⣠⠿⣛⣩⠶⠚⠋⠉⠤⠀⣸⣿⡉⠁⣀⣀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠓⣿⠉⣡⣶⣿⣿⣿⣿⢗⣿⣿⣿⣿⣷⣶⣧⣄⠀⠀⠻
⠀⠀⠀⠀⠀⢀⣿⠀⣹⡿⠿⣛⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⣿⣿⣷⡶⣾⣿⣿⣿⣿⣿⣿⡿⣯⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⢿⣿⡟⠋⠉⠉⣱⡋⣸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⡴⠋⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠖⠓
⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⡀⠋⠁⠀⠀⠀⣠⠰⠚⠋⠉⠹⠱⢦⢄
⠀⠀⠀⠠⡀⠀⠷⠇⠉⡆⠀⠀⠀⠀⠀⠀⠀⠠⠖⠒⠂⠈⠀⠀⠀⠀⠀⡼⡙⡄⠁⠃⠀⠀⠀⠂⠞⣡
⠀⠀⠀⠀⠀⡤⣀⠠⠾⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠗⠀⠀⠀⠀⠀⠀⠀⠈⠘⢼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠉⠂⠀⢀⣀⠀⠃⠀⠀⠀⠀⠓⢤⣖⠀⢀⣀⣀⠀⣀⠴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣧⣼⡼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠋⠉⠙⠓⠦⣄⣠⠴⠒⠒⠛⠒⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠉⠉⠑⢎⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠒⠛⠛⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⢀⣀⣤⠤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣀⡀
⢀⡴⠊⠉⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠉⠉⠉⠑⢦⡀
⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⢳
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢹⠀⠀⠀⠀⡇⠀⠀⠀⢠⠇⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠸⡆⠀⠀⠀⡇⠀⠀⠀⣼⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⣇⠀⠀⠀⡇⠀⠀⢀⡇⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣸⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⣆⠀⠀⠘⡆⠀⠀⡇⠀⠀⡏⠀⠀⢠⠃⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠸⡀⠀⠀⣇⠀⠀⡇⠀⢸⠁⠀⢠⠇⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⢧⠀⠀⢹⠀⠀⡇⠀⡞⠀⢀⡎⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠘⡆⠀⢸⠀⠀⡇⢀⡇⠀⡞⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⢱⠀⠈⡇⠀⡇⢸⠁⡼⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠛⠓⠒⠓⠒⠓⠚⠒⠓⠛⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣤⣀⣀⣀⣀⣀⣠⡤⠴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡰⠃⡠⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠠⡶⡒⠦⣼⡀⠘⣆⠜⠊⠉⠉⠒⠦⡀⠀⠀⢀⡤⠂⠉⠀⠀⠀⠙⢶⠁⢀⡼⠥⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⠢⡀⠑⣤⠃⠀⠀⠀⠀⠀⠀⠈⢆⢀⠎⠀⠀⠀⠀⠀⠀⠀⠈⡗⢁⠔⠋⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⣈⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣋⣀⣴⣃⣀⣀⣀⣀⣀⡀
⠀⢠⠄⠒⠒⠂⠉⠉⠉⠁⠀⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⣀⡭⠟
⠀⠀⠈⠑⠒⠂⠤⣄⠀⠀⢰⡇⠙⠓⠠⠤⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠤⠚⠋⠁⢸⠀⠀⠀⢠⡴⠚⠉
⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠸⡇⠀⠀⠀⠐⢲⠖⠒⠀⡤⠄⠀⡀⠀⣠⠀⠠⢤⠤⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉⢑⣲⡄
⠀⠀⠀⠻⠖⠒⠒⠒⠒⠒⠺⡇⠀⠀⠀⠀⢸⠀⠀⠀⣇⡀⠀⠘⢦⡃⠀⠀⢸⠀⠀⠀⠀⠀⠀⣸⠒⠒⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⣀⠀⠘⠀⠀⠀⠧⠤⠀⠠⠃⠑⠄⠀⠘⠀⠀⣀⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠝⠋⡰⠒⣒⠂⠠⠤⠤⠤⠤⠐⣲⠒⠺⣍⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣠⠊⠀⠀⡸⠓⢤⡀⠀⠀⡠⠞⠙⢄⡀⠈⢣⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⢃⡠⠔⠈⠀⠀⠀⠈⠓⠊⠀⠀⠀⠀⠉⠓⠢⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢀⣶⠾⢶⣦⠀⢀⣴⠾⠷⣦⡀⠀⣶⡶⠶⣶⣄⠀⠀⣶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢸⡇⠀⠀⣿⠀⢸⡏⠀⠀⢹⡇⠀⣿⡇⠀⠈⣿⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠆⠀⠀⣸⡇⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⢽⡇⠀⣿⡇⠀⠀⣿⡇⠀⣿⣀⣀⡀⠀⢦⣄⡀
⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⢀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣸⡇⠀⣿⡇⠀⠀⣿⡇⠀⣿⠉⠉⠉⠀⠀⠉⠛⢶⣤⡀
⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣤⡀⢸⡇⠀⠀⣼⡇⠀⣿⡇⠀⠀⣿⠇⠀⣿⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠆⠀⣿⠀⠀⠀⢸⣧⡀⣀⣿⠀⢸⣧⡀⢀⣾⠇⠀⣿⣇⢀⣰⡿⠀⠀⣿⣀⣀⣀⠀⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠉⠛⠛⠉⠀⠀⠙⠛⠛⠉⠀⠀⠀⠛⠛⠛⠋








⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠂⢀⣭⣍⢉⣭⣭⣭⣭⣭⡀⠐⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠃⢸⡇⠘⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⠄⠀⠃⢸⡇⠘⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⡀⢸⡇⢈⠁⠠⠶⠦⠠⠶⠶⠶⠶⠶⠄⠈⡁⢸⡇⢈⣁⣈⣁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠁⢸⡇⢈⡀⠐⠶⠶⠶⠶⠶⠶⠶⠶⠂⢀⡁⢸⡇⠈⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⢠⡀⠐⠛⠓⠐⠛⠛⠛⠛⠛⠂⢀⡄⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠠⠄⠀⣀⣴⠄⠀⣾⠠⣦⣀⠀⠠⠄⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠰⠄⠀⠻⣦⡀⣸⠇⢀⣴⠟⠀⠠⠆⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠰⠂⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠐⠆⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛





⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣿⣟⣛⣿⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠄⠀⠀⠀⠔⠃⠀⠑⠄⠀⠀⢤⢤⠄
⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠸⣿⣿⣖⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢿⣿⣿⣿⡶⢀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣯⡶⣫⠄⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⡈⠙⡿⣅⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⡶⡰⣀⡀⢀⠀⣀⠀⠰⣄⣀⠱⠗⠆⠾⢦⣬⣼⣣⣡⡑⢮⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠚⠛⠓⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⢦⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣕⣺⣏⣂⣿⣑⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⢟⣿⠿⠿⠿⠿⠿⢿⡿⠿⣿⣟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣭⣭⣭⡎⣿⣷⣜⣿⣛⣛⣛⣛⣻⣟⣛⣿⣻⣟⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⠿⠿⠿⠿⢃⢹⣿⠟⣿⣿⣿⢻⣿⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣸⣿⣿⣿⣿⣿⡆⣸⣷⣜⣿⣿⣱⣿⣿⣣⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⢣⣿⣿⡿⣿⣿⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣙⡍⣶⣿⣿⣯⡘⣿⡿⣣⣿⣿⢃⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣛⠿⣣⣦⣼⣍⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣚⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⡟⡛⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⢸⡆⠀⠀⠀⣿⣿⣿⡇⢤
⠀⠀⠀⠀⠀⠀⠀⢸⡟⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡘⣿⠀⠀⠀⣿⣿⣿⡇⠼
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⣂⠠⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠵⢻⠆⠁⣸⠇⠀⠀⠀⠀⠀⠀⡀⡀⢀⣀⣬⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡫⠉⢁⡭⡝⡆⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢈⣿⠟⠉⠋⠙⠻⣾⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⡀⠀⠀⠀⣨⠟⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠒⠲⢤⣾⠃⠀⠀⠀⠀⠀⠀⠀⣾⡅⠀⠰⡀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣩⡄⣀⠲⢠⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠭⠖⠁⠀⠀⠀⠀⠰⡞⠉⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣦⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣰⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠰⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⣠⣾⣿⣴⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣿⣿⣿⣿⣿⣿⣿⡿⠻⢿⣿⡁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠾⠛⣻⣿⣿⡿⠿⠋⠋⠀⠀⠀⠀⢻⡉⠀⠀⠀⠀⠀⢿⣹⣇⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠱⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠁⠀⠀⡀⠀⠢⡈⠁⠀⢢⠂⠀⠀⠀⠀⠀⢿⡿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡰⣱⡀⠀⠀⠀⠀⠀⢸⠟⢻⠷⡀⠀⠑⠀⠀⠀⠈⠄⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡅⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠤⠂⠀⠠⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠈⠒⠆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠜⠃⢶⠦⢄⠀⠁
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠉⠉⠉⠁⢹⣿⣿⠄⠀⠀⠀⠀⠸⣴⡢⢀⣼⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⢀⡀⠀⠀⠀⢸⣿⣿⠀⠀⠘⣅⠁⠃⠘⡆⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣛⣿⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠀⣿⠎⠈⡀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠁⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠘⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠨⠀⠀⠀⠀⠀⠒⠐⠀⠀⠐⠒⠒⠒⠒⠒⠒⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⣹⢛⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡯⠿⠞⠛⢚⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠋⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⢀⠀⠀⢠⣀⠀⢠⣿⡟⠀⠀⠀⠀⡀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠈⠿⠛⠓⡤⠼⣿⣿⣶⣾⣦⡀⠀⠔⠪⠑⠀⠀⠀⠐⣀⠀⡀⠀⣀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣄⠈⠀⠈⠈⠁⣾⣿⣆⠀⠀⠀⣦⣿⣶⣾⣿⣿⣿⣷⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⡀⠠⣄⣀⣀⠀⠀⠀⡘⢿⣶⡊⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠸⠿⢿⣿⡷⢄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠢⣄⠈⠙⠛⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂⠀⠀⠀⠀⠈⠂⠀⠀⠏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠶⠀⡀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠁⣤⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣼⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣾⣿⣱⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⡿⣡⣿⣿⣿⣿⣿⣿⠋⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣿⡿⣱⣿⣿⣿⣿⣿⣿⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⣿⡿⣱⣿⣿⣿⣿⣿⠟⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣰⣿⣿⣿⣿⣿⠏⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣻⣿⣿⣿⣿⡟⠑⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣿⣿⣿⣿⡟⠁⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢏⣿⣿⣿⣿⡿⡇⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⣼⣿⣿⣿⣿⡫⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣼⣿⣿⣿⣿⡟⠂⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢟⣾⣿⣿⣿⣿⣿⣶⠶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠇⠀⠀⠀⠀⠀⠀⠀⣾⡿⢫⣾⣿⣿⣿⣿⣿⡿⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡎⠀⠀⠀⠀⠀⢀⣾⣿⣵⣿⣿⣿⠿⠿⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣠⣐⠂⠀⠀⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣿⠆⠀⠀⣿⣿⣿⡿⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⠘⢿⡀⠂⠀⠀⠙⣷⠀⠀⠀⢀⡎⠀⠀⠀⠄⠀⢄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠬⠇⠀⠀⠀⠀⠘⡀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠽⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⣿⣒⣫⣽⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣿⠻⠋⠋⢁⣫⣾⣿⡤⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⡒⠤⠊⠐⠽⠋⣀⣿⡀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠐⢯⣭⠉⠉⠀⠈⠁⣀⣠⡀⢀⡿⠋⠙⠻⠗⠚⠁⢀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⢿⢞⣤⢀⣠⠞⣁⣬⡽⠟⠀⠀⠀⠀⠀⠀⣦⣿⠀⠹⣄
⠀⠀⠀⠀⠀⢴⠊⠙⠛⠒⠚⠋⠁⣠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠉⠓⠒⠒⠛⢦
⠀⠀⠀⢀⠤⠬⢿⣾⣷⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠐⣄⢦⣸⣿⣏⢆⢸⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠈⠳⣻⣿⣿⢸⡾⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡅⠀⠒⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠀⢀⣀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠐⣱⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠌⣧⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡠⣜⣧⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡽⣎⢦⠀⠀⠀⠀⠀⠀⡀⠀⢰⡿⠠⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡝⣌⢆⠀⠐⠀⡄⠀⠀⠀⣸⠏⠀⠀⠀⠂⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣞⡌⡄⢸⡇⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢱⣸⡇⠀⠀⡰⣹⢃⠀⠀⠀⠀⢀⣠⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣯⡇⠘⢠⢃⡟⠀⠀⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⢻⠇⡧⡐⣼⠃⠀⠀⠀⠀⠘⠦⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠳⢀⣇⡀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡜⠀⠹⣄⣀⣀⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⡆⠐⡇⠀⢸⠉⢱⠔⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢒⠁⠁⠀⠇⠀⠇⠀⢸⠀⡸⠀⠘⠀⠘⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⡅⠘⡀⠘⠀⢰⠀⠀⠀⠀⠀⡇⠀⠄⢀⠀⠀⠈⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠺⡀⢱⠀⢣⠀⠆⠀⡄⢸⠀⡆⠀⠃⠀⠀⡌⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠡⠀⠆⠀⠄⢀⠀⢁⠘⠀⡇⢠⠀⡀⠀⠀⠠⠀⡐⢀⠊⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠂⠀⢀⠘⡀⠐⠀⡄⠀⠀⡆⡇⠘⠀⠁⠀⢠⠁⡐⠠⠃⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⢆⠀⠀⠱⡀⠂⢠⠀⠄⡇⠁⠀⠐⢰⠀⠆⡐⡠⠃⡔⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢂⠀⠀⢣⠘⡀⡀⡃⠀⠀⠀⠀⡆⡈⡰⡰⢁⠎⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠄⢂⢁⠀⢰⠠⠀⠇⢠⠰⠡⡑⡡⢃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠱⡈⠌⠆⠀⠀⠀⠀⠀⠀⠂⠚⡐⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣔⠜⠌⡄⠀⢀⣠⡆⠀⠀⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣀⣑⣼⣞⣸⣼⣿⣧⣮⠚⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡄⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀
⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⡇⠀⠀⠿⠿⠀⢀⣀⣸⣿⣀⣀⠀⢀⣀⣿⣿⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⣾⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⢸⣿⡇⠀⠀⣠⣄⠀⠘⠛⢿⣿⠟⠛⠀⠘⠟⣿⣿⠟⠛⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣴⣿⡟⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢀⣀⡀⠀⠀⣀⣀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠛⣿⣷⠀⢸⣿⡇⠀⠀⣿⣿⠀⢸⣿⡟⠛⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣴⣴⣿⡟⠀⢸⣿⣧⣤⣤⣿⣿⠀⢸⣿⣧⣤⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠁⠀⠀⢸⣿⡏⠉⠉⣿⣿⠀⢸⣿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣀⠀⢀⣀⣀⣀⣀⣀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠘⠿⢿⣿⠿⠟⠀⢸⣿⣷⣄⣠⣾⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡟⢿⡿⠛⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠉⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⡇⠀⣿⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿
⠙⠻⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⣿⣿⠟⠟⠛⢿⣿⡟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⡆
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣗⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠼⣿⡿⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣠⣿⣷⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⢠⣶⠺⠏⠀⢻⣷⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠉⣉⡀⠀⠉⢩⣿⣶⣶⣄⣀⠘⠃⠀⠉⠀⠀⠀⠀⡟⠿⣿⣾⣥⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢠⣿⣷⣶⣶⣿⣿⣷⣤⣾⣿⡿⠯⠄⠙⡄⠀⠀⠀⠀⠀⠀⠐⠃⠀⠘⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⣿⡿⠛⠛⣛⣋⡉⢩⣽⡿⣿⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢻⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⣿⠋⠀⠀⢰⣿⣿⣷⣿⣿⣷⡙⠛⠻⢿⣦⡄⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠚⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣎⣿⠴⠖⠀⠀⠈⠉⣉⣭⣿⣿⣶⣤⠀⠀⠀⣿⣷⡆⠀⠀⠀⠀⠉⠀⠀⢀⠄⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⢌⣉⣤⡾⠁⠀⠀⠀⠙⠿⠿⠿⠟⠛⠉⠀⣴⣜⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⣴⢾⣾⠃
⠀⠀⠀⠀⣀⠀⠀⠠⠀⢀⣀⠀⠁⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡄⢀⣉⡙⢿⣿⣿⡄⠀⠀⠀⠀⢀⠀⣼⡏⢸⡇
⠀⠀⠀⠀⠀⣄⡉⠙⡄⠀⠉⢻⣿⣶⠿⠀⠀⠻⣿⣿⣿⣿⣿⣏⠀⠈⠻⡇⣸⣷⣿⣿⡀⠀⠀⠀⠀⢠⣿⡇⢸⣷⡀
⠀⠀⠀⠀⠀⢈⣄⠀⣱⡀⠈⡛⣿⣇⡀⡀⠀⣠⣿⣿⣿⣿⣿⣿⠞⠂⠴⡄⢸⣿⣿⣿⣷⠀⠀⠀⣀⣾⡟⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠉⠀⠈⡏⠀⠈⢹⣿⠁⠀⠀⠀⣽⣿⣿⣿⣿⣿⣤⣴⣦⡇⣿⣿⣿⣿⠇⠀⠀⠀⢠⡿⠀⠀⠀⢸⣿⣅
⠀⠀⠀⠀⠀⠀⠀⣿⠘⢿⠂⠀⠺⣿⣿⡀⠀⠀⣬⣿⣿⣿⣿⡏⢀⣼⣿⡆⢹⣿⣿⣿⣿⣧⠀⠀⠀⣡⠀⠀⠀⠈⣍⣉
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠲⣇⠀⠸⣿⣿⡇⠀⠀⢈⣿⣿⣿⣿⣇⡈⢸⣿⠀⠉⣿⣿⣿⢽⣿⡇⠀⠀⢸⡆⠀⠀⠀⣏⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢸⡄⠀⢾⣿⣿⠀⠀⠀⣹⣿⣿⣿⡟⢃⣸⡇⠀⠈⣿⣿⡇⢰⣿⡇⠀⠀⢸⠃⠀⠀⠀⢡⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢤⠀⠀⢻⣿⣇⠀⠀⠙⣿⣿⣿⡧⠈⠁⠀⠀⢸⣿⣿⡅⣼⣿⡇⠀⠀⣿⠀⠀⠀⠀⠉⣁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠁⠀⠸⢿⣿⠁⠀⠀⣿⣿⣿⣷⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⣼⡏⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠀⠘⣿⡀⠀⠀⠹⣿⣿⡏⠁⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⢠⣿⠁⠀⠀⠀⢠⡞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠄⠀⢠⣿⣿⣷⠀⠀⠀⢀⣿⣿⣿⣿⣿⠀⣴⣿⡇⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠘⠂⠀⠘⣻⡀⠀⠈⣿⣿⣯⡄⠀⠀⢸⣿⣿⡿⡟⠁⢺⣿⠟⠀⠀⠀⡠⠟⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠹⣇⠀⠀⣼⣿⡟⠀⠀⢠⣿⣿⣿⡶⠓⠀⣾⡟⠀⠀⠀⠀⣤⣤⣬⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢹⠁⠰⣿⣿⡧⠀⠀⣾⣿⣿⣿⠖⠀⣼⠿⠀⠀⠀⠀⠘⠛⠋⠁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣿⠆⠀⢿⣿⡁⠀⣸⣿⣿⡟⠃⠀⣼⠋⠀⠀⠀⠀⢀⣤⠂⠀⠀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⡀⠀⠀⠀⠀⢸⣿⡟⣰⣿⡿⠯⠀⣠⡾⠃⠀⠀⠀⠀⠀⣸⣿⡄⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⡀⠀⠀⠀⢸⡟⣱⣿⠟⠃⢠⣴⠟⠀⠀⠀⠀⠀⠀⢺⣶⣅⠀⠉⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠰⣿⣿⠁⠀⣰⡿⠁⠀⠀⠀⠀⣴⣴⢶⡛⠃⢠⣈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⣸⣿⣿⡿⢿⣿⡆⠀⠀⠀⡜⠛⢄⠙⣦⠿⠒⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢀⡿⣿⣿⣦⡶⠿⠇⠀⢠⠀⣿⡄⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⢛⣿⢻⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠲⣿⠉⠉⠉⠙⣿⠓⠒⠲⠤⣄⣀⡀
⠀⠀⠀⠀⠀⣠⠖⠛⠻⣧⠀⠀⠀⠀⠘⣇⠀⠀⠀⢸⡄⠀⠀⠀⢸⡟⠉⠛⢦⡀
⠀⠀⠀⡔⠚⢻⡀⠀⠀⠈⣧⡀⠀⠀⠀⢻⡀⠀⠀⠈⣧⠀⠀⠀⢸⠇⠀⠀⠀⢹⣦⣀⡀
⠀⠀⡜⢧⠀⠈⢷⡀⠀⠀⠘⣷⠀⠀⠀⠈⡇⠀⠀⠀⣿⠀⠀⠀⡿⠀⠀⠀⢀⡾⠁⠈⠙⢦⡀
⠀⠀⡇⠘⣧⠀⠘⣧⠀⠀⠀⢹⣇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢀⡞⠀⠀⠀⠀⠀⢹⡆
⠀⠀⡇⠀⠸⣆⠀⢹⡆⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢠⡇⠀⠀⣿⠀⠀⢠⡟⠀⠀⠀⠀⢀⣴⡾⠻⣄
⠀⠀⢹⡀⠀⢹⡄⠈⣷⠀⠀⠀⢹⡆⠀⠀⡇⠀⠀⡾⠀⠀⣸⠃⠀⢠⡟⠀⠀⠀⢀⡴⠛⠁⠀⠀⠈⢷
⠀⠀⠀⢧⠀⠀⢷⠀⢹⠀⠀⠀⢸⠃⠀⢸⠃⠀⣸⠃⠀⣰⠏⠀⣠⡟⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠈⣇⠀⠘⡇⢸⡇⠀⠀⣾⠀⠀⡏⠀⣰⠃⠀⣰⠏⠀⣰⠏⠀⠀⣠⠞⠁⠀⠀⠀⢀⣠⠶⠛⠉⢳⡀
⠀⠀⠀⠀⠘⣆⠀⢹⢸⡇⠀⢀⡇⠀⣸⠁⢰⠃⠀⣰⠃⠀⣰⠏⠀⢠⡞⠁⠀⠀⢀⣠⡶⠛⠁⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠘⡆⠘⣾⡇⠀⢸⠃⢠⡇⢠⠏⠀⡴⠃⠀⣼⠏⠀⣰⠋⠀⠀⢀⣴⠟⠉⠀⠀⠀⢀⣀⣤⡶⢿
⠀⠀⠀⠀⠀⠀⢹⡄⣿⡇⠀⡾⠀⡼⢠⠏⢀⡜⠁⢀⣼⠃⢀⡼⠁⠀⢀⣴⠟⠁⠀⢀⣠⠴⠞⠋⠉⠀⠀⢸
⠀⠀⠀⠀⢀⣠⠞⢳⢹⡇⢠⡇⢰⢃⠏⢠⠞⠀⢀⡾⠁⢠⠞⠀⢀⡴⠋⠀⢀⡤⠞⠋⠀⠀⣀⣀⣤⡴⢶⡟
⠀⠀⠀⠀⠺⡁⠀⠈⣿⡇⢸⢃⣇⡞⣰⠋⠀⣠⠟⢀⡴⠃⠀⣰⠟⠁⡠⠞⠉⢀⣠⡴⠞⠛⠉⠁⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠙⢦⠀⢻⡇⢸⣼⡜⡼⠁⢀⡴⠋⣠⠟⠁⣠⠞⢁⡴⠋⣀⡴⠞⠋⠁⣀⣠⣤⣤⡴⠶⣶⠟
⠀⠀⠀⠀⠀⠀⠈⠳⣼⣿⢸⣿⡿⠁⣠⠟⣠⠞⠁⣠⠞⣡⠖⣩⠔⣋⡥⠴⠒⠛⠉⠉⠀⠀⠀⢀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣣⢞⡥⠞⣁⡴⢊⣴⡾⣱⣽⣶⣭⠤⠴⠶⠶⠒⠒⠚⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣏⣥⣖⣪⣵⡾⠿⠟⠋⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠒⠤⠤⣀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠛⠀⠘⠃⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠷⠶⠶⠶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠠⣤⣤⠀⣤⣤⣤⡄⠠⣤⡄⠀⣤⡄⠀⣤⣤⣤⡄⢀⣤⣤⣤⡀⢀⣤⣤⣤⡀⢀⣤⣤⣤⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⣿⠀⢹⡇⠀⢸⡇⠀⢸⡇⠀⣿⠀⢹⡇⢸⣿⠀⣿⡇⢸⣿⠀⣿⡇⢸⡏⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠸⡿⠀⠿⡶⠿⠇⠀⠸⠇⠀⠸⠇⠀⠿⠶⠿⠃⠘⠿⠶⠿⠃⠘⠿⠶⠿⠃⠸⠿⠶⠿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣀⣀⠀⢀⣀⡀⠀⣀⣀⣀⠀⠀⣀⣀⣀⠀⢀⣀⡀⠀⣀⣀⣀⠀⠀⣀⣀⡀⠀⣀⣀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠛⢻⡇⠈⢻⡇⢐⣿⠛⣿⡇⢸⣿⠛⣿⡆⠘⣿⡇⢸⣿⠛⣿⠄⢸⡟⠛⣿⠀⠙⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⣿⣤⣼⡇⠀⢸⡇⠸⣿⣤⣿⡇⠸⣿⣤⣿⠇⠀⣿⡇⢸⣿⣤⣿⠂⢸⣧⣤⣿⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣈⡉⠉⢀⣀⣈⠁⠀⣈⡉⠉⣀⡀⠈⠉⣁⣀⡀⠉⢁⣀⠉⠉⣁⣀⡀⠉⢉⣁⠀⢀⣉⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⢻⣿⠀⣿⠛⢻⡇⠈⢻⡇⠀⠻⠇⠀⠸⠟⠛⠿⠀⠘⠿⠀⢸⡟⠛⣿⠀⠙⣿⠀⠙⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⣿⣤⣼⡇⠀⢸⡇⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣄⠀⢸⣧⣤⣿⠀⠀⣿⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢈⣁⡀⠈⠉⢉⣀⡀⠈⠁⢸⣿⠀⠀⢴⣤⣄⠀⠀⠀⣿⡇⠀⣉⡉⠁⣀⡀⠉⠀⣀⣉⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠛⢿⡇⢰⣿⠛⣿⡆⠀⠸⣿⣄⣀⠘⠿⢿⡄⢀⣠⣿⠇⠀⢻⣿⠀⢻⣿⠀⣾⡟⢻⡷⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠨⣿⣄⣼⡇⠸⣿⣀⣿⡇⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⢸⣿⠀⢸⣿⠀⣿⣇⣸⡧⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠉⢉⡉⠀⠀⡉⠉⡉⠀⠀⠀⠀⣸⡇⠀⢸⡇⠀⢸⣇⠀⢀⠀⠈⢉⠀⠈⠉⡀⠈⠉⡉⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⢀⣿⡛⠛⠃⠀⢸⡇⠀⠘⠛⢛⣿⡀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⣿⡄⠀⢀⣼⣧⡀⠀⢰⣿⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠛⠀⠀⢿⣧⣼⡟⠀⠀⠛⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢺⣇⠀⠀⣼⠀⠀⠀⣴⠀⠠⣄
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⡏⠀⠀⡿⠀⠀⣸⠃⠀⠀⢈⡳
⠀⠀⠀⠀⠀⡦⠭⠀⠀⠀⠀⠀⠀⢀⣠⠊⠁⢀⠇⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⡇⠀⢠⠇⠀⣰⠏⠀⢀⡴⠋⠀⠈⢳⡀
⠀⠀⠀⠀⠀⢳⡈⠀⠀⠠⣤⣿⣿⣾⣿⡶⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠁⠀⡞⠀⡰⠃⢀⠔⠉⠀⠀⠀⣀⣀⢗
⠀⠀⠀⠀⠀⠈⢣⣀⣈⣙⣟⣛⣿⣿⣿⣥⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣾⣶⣞⣧⠜⠁⠀⠁⣀⠤⠔⠚⠋⠉⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠟⠋⠋⠙⠋⠉⠀⠀⠀⠀⡀⠀⠀⣤⣶⣾⣿⣿⣿⣿⣵⣿⢧⠒⠉⣀⣀⣀⣀⠤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢿⡛⠻⠻⢿⣿⣿⣿⣿⣁⣁⣀⣀⣀⣀⡀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⠻⠆⠀⠀⠀⠁⠐⡀⠀⠀⠘⣦⣀⠀⢸⣿⣿⣿⣿⣆⣀⡀⠈⠉⠉⠙⠛⢷⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⠂⠀⠀⠈⠻⠿⡿⣿⣿⠛⣿⣿⣿⣭⣟⡓⠶⣤⣄⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣖⡛⢙⣮⡿⣇⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠉⠁⡴⠟⠉⠉⠙⠙⠛⠳⣤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣿⣿⡇⠐⠲⠊⢉⢴⢿⣶⠀⣀⡀⠀⠀⠀⡔⠒⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢫⣿⣯⣔⠁⠀⠀⡠⠀⠎⠡⠁⡏⣷⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣃⡾⣿⣿⡆⠀⠀⢀⠌⡀⠀⠀⣴⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⡼⠹⢟⣾⣿⣴⣮⣰⣀⠀⣀⣿⣾⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡈⢀⡟⠻⣿⣿⣿⣿⢿⡏⢿⠛⢁⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣾⣤⣸⠀⣌⠀⠁⢀⡇⢀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⢿⣿⣷⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠁

⣶⣶⡶⡶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣦⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀⣴⣾⠟⠛⠉⠉⠙⠛⢿⡿⠟⠛⠉⠙⠻⢿⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠛⠉⠛⠛⢿⠟⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⡟⠋⠉⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡿⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠹⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠋⢻⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡇⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠟⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡗⠀⠀⠀⠀⡄⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⢀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡇⠀⠀⠀⠐⠀⠀⠀⠀⠸⢷⣄
⠀⠀⠀⢀⣴⠿⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠁⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢹
⠀⠀⣰⡿⠁⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⠀⠀⠀⡞⠀⠀⠀⡐⠀⠀⠀⣼
⠀⠀⣿⡇⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣇⠀⠀⠀⣿⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢻⠀⠀⠀⢿⠀⠀⣸
⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠸⡇⠀⠀⢸⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⣾⡟⠙⢧⡀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢻⠀⠀⢸⡆⠀⠃⠀⠀⠀⠀⠀⠀⢠⡞⠁
⢸⣿⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠘⡀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠘⣿⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠹⣷⡀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠈⢄⠀⠀⠑⠀⠀⡀⠀⠰⠀⠀⡜
⠀⠀⣿⠿⢦⡀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢦⡀⠀⢳⠀⢡⠀⠀⠀⡌
⠀⠀⣿⠀⠀⠙⠦⣀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠈⠳⣄⠀⠀⠑⡄⠀⠃⠀⠃⠀⠠
⠀⠀⢻⡇⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠈⠓⠦⡀⠀⠀⠈⠑⢄⠀⠀⠠⡀⠰⡀
⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠙⠳⢄⣀⠀⠀⠀⠈⠙⠲⣤⣀⠀⠀⠀⠀⠈⠓⠦⢄⡀⠀⠀⠀⠀⠈⠓⠂⢀⡀⠀⠀⠠⡀⠑⢄
⠀⠀⠀⠀⠈⢳⡦⣄⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠉⠓⠲⢤⣀⡀⠀⠀⠀⠉⠛⠂⠤⣄⣀⠀⠀⠀⠉⠒⠄⠀⠀⠑⠀⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠈⣧⠀⠉⠒⠦⣄⡀⠀⠀⠀⠈⠉⠓⠶⣤⣀⠀⠀⠀⠀⠉⠑⠲⠦⢤⣄⣀⡀⠀⠈⠉⠀⠀⠀⠤⠀⣀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠉⠛⠲⢦⣄⣀⠀⠀⠀⠈⠙⠓⠶⠦⣤⣄⣀⡀⠀⠀⠀⠉⠉⠉⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣶⠦⠤⢤⣤⣀⣉⣉⣓⠲⢦⣤⣄⣀⣀⡀⠀⠈⠉⠉⠉⠛⠒⠒⠒⠒⠒⠒⢂⣉⣉⡠⠤⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠍⠉⠛⠛⠛⠛⠛⠓⠒⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠐⠂⠀⠀⠀⠂⠀⠂⠂⠀⠂⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠄⠀⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⢻⡋⠛⣛⣛⣒⡢⢤⣀
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⡀
⠀⠀⠀⠀⣜⣀⠊⢗⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢌⡱⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢈⣾⣹⣿⣶⣤⣁⢋⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠈⠑⠢⡤⠤⠀⢀
⠀⠀⣤⠒⠨⡙⠻⢾⣿⢏⠂⡀⢀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠈⠑⢂⠀⠀⣀⢀⠀⢸⡾⠀⠀⠀⠠⠀⠀⠀⣸⡆
⠀⠜⡈⠀⠴⣠⢮⠕⢀⠀⠉⣸⠩⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⣑⠈⢸⡇⠀⠀⠀⡄⠀⠀⢠⣿⠁
⠀⠑⠚⠵⣢⠄⡉⢀⠂⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠛⠉⠙⠛⠓⠳⠶⣾⡇
⠀⠀⠀⠀⠀⠉⠒⠂⠀⠀⠀⠀⠀⠀⠴⠶⡈⢦⠀⠀⠠⠀⢀⡠⠊⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣒⣒⡠⡞⠊⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠸⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡴⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣆⣾⣀⣸⣆⣆⣀⣀⣀⣀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⠖⠚⠛⠓⠶⣶⣤⡤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡖⠋⠉⠀⠀⠈⢻⣆⠀⠀⠀⠀⠸⣿⠀⠀⠀⠉⠉⢳⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢸⣿⠁⠀⠉⠻⣦⡀
⠀⠀⠀⢀⡴⠋⠙⣧⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⣦⣀
⠀⠀⠀⣼⣆⠀⠀⠹⣦⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣠⡿⠁⠉⠻⣦⡀
⠀⠀⢸⠀⠹⣆⠀⠀⢹⣇⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢰⡏⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⢸⠀⠀⢻⡄⠀⠀⢿⡄⠀⠀⠀⠀⢿⡆⠀⠀⠀⢸⡇⠀⠀⠀⢀⡏⠀⠀⠀⣾⠃⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣸⣷
⠀⠀⢸⡆⠀⠀⢿⡀⠀⠸⣷⠀⠀⠀⠀⢸⣧⠀⠀⠀⢸⡇⠀⠀⠀⣸⠇⠀⠀⢸⡟⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⢷⡀
⠀⠀⠀⣷⠀⠀⠘⣷⠀⠀⢿⡆⠀⠀⠀⠈⣿⠀⠀⠀⢸⠇⠀⠀⢠⡟⠀⠀⢀⣿⠁⠀⠀⣼⠏⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠸⣇⠀⠀⠸⣇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣾⠀⠀⠀⡾⠁⠀⠀⣾⠃⠀⠀⣼⠏⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⢹⡄⠀⠀⢻⡄⠈⣿⠀⠀⠀⠀⣿⠀⠀⢠⡇⠀⠀⣼⠃⠀⠀⣼⠃⠀⢀⣾⠏⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣠⣶⢿⣇
⠀⠀⠀⠀⠀⢻⡄⠀⠈⣷⠀⣿⠀⠀⠀⢸⡇⠀⠀⣾⠀⠀⣸⠃⠀⢀⣼⠃⠀⢀⣾⠃⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢿⡀⠀⢸⡆⣿⠀⠀⠀⣸⠁⠀⢰⠇⠀⣴⠃⠀⢀⡾⠁⠀⢠⣾⠃⠀⠀⣴⠟⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⣇⣿⠀⠀⠀⡿⠀⢀⡟⠀⣰⠃⠀⢀⡾⠁⠀⢠⡿⠃⠀⢠⡾⠁⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢿⣿⠀⠀⢸⡇⠀⣸⠃⣰⠏⠀⢠⡟⠀⠀⣰⡿⠁⠀⣰⠟⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⣀⣤⣴⠾⠟⠋⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⢸⣿⠀⠀⣼⠀⢠⡏⢠⠏⠀⣰⠏⠀⠀⣰⡟⠀⢀⡼⠃⠀⠀⢀⣴⠟⠁⠀⠀⢀⣠⡶⠟⠋⠁⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⣠⡶⠛⢻⡘⣿⠀⢀⡟⠀⡾⢠⠏⢀⡼⠃⠀⠀⣼⠏⠀⢠⡞⠁⠀⢀⣴⠟⠁⠀⢀⣤⠞⠋⠁⠀⠀⠀⣀⣠⣤⣶⠶⠿⣿
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣧⣿⠀⢸⡇⣸⢁⡟⢠⡞⠁⠀⢀⡾⠃⢀⣴⠏⠀⠀⣴⠟⠁⠀⣠⠞⠋⠀⠀⣀⣤⡶⠞⠛⠉⠉⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢸⣿⡀⢸⢃⣏⡞⣠⠏⠀⠀⣰⠟⠁⣠⡟⠁⠀⣠⡾⠁⢀⡴⠋⠁⣀⣤⠶⠟⠉⠁⠀⠀⠀⣀⣀⣀⣠⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠸⣿⡇⢸⣿⣸⣰⠏⠀⢀⡾⠋⢀⡼⠋⠀⣠⡾⠋⣠⠶⠋⣠⡴⠞⠋⢀⣀⣤⣤⠶⠶⠿⠛⠛⠛⠉⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣿⣿⢸⣿⣧⠏⠀⣴⠋⢀⡴⠟⠁⣠⠞⢁⣤⠞⣡⠴⣋⣡⠴⠖⠛⠉⠉⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡟⣠⢞⣡⠾⠋⣀⡴⠚⣡⣴⡯⢖⣯⣶⣋⣥⣤⡤⠶⠶⠶⠶⠒⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠛⣋⣠⠴⢚⣩⣴⣿⣯⠷⠞⠛⢿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣟⣛⣋⣉⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠦⢾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⡔⠁⠀⠲⣤⣀⣀⣀⣠⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡋⠉⠀⠀⠘⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣠⠏⠞⠀⠀⠀⠀⠀⢹⣿⣿⣿⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣦⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣶⣤⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⢣⡻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡮⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⢿⢿⣿⣯⣻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠳⣝⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⢀⠤⢖⠉⢂⠁⠀⠀⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠡⡀⠈⠻⣷⣍⠿⣿⣮⣛⣿⣷⡲⡄
⠀⠀⠀⠀⠒⠈⠁⠂⠑⠂⣠⢖⡵⠁⠀⠠⠂⡔⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⡄⢀⠊⢿⣷⣬⡻⣿⣷⣯⣟⠷⣄
⠀⠀⠀⢠⠠⠴⠊⣀⡴⠚⠑⠁⣠⠞⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡑⢄⠉⠻⣷⣄⡙⠿⣿⣿⣶⠄
⠀⠀⠀⠀⣀⣤⣾⡿⠟⠁⣠⡾⠁⠀⢠⠊⠀⠀⠀⠀⠀⢰⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠙⢶⡳⣄⠙⠻⣿⣶⣌⠻⣿⣆
⠀⠀⠀⢹⣿⠟⠉⠀⣠⣾⢟⠁⠁⣠⠃⠀⠀⠰⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠱⣄⠀⠙⢾⣷⣄⠈⠻⣿⣿⣾⠟
⠀⠀⠀⠂⠀⢀⣤⣾⣿⠟⠁⢀⡴⢁⠄⠀⡠⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠀⠂⠐⠄⠐⠆⠺⠶⠀⢢⡹⣿⣷⣆⣌⠻⢷⡄
⠀⠀⠀⠰⣾⣿⣿⠟⠁⠀⢠⠎⠠⠂⠀⣰⠇⠀⠀⢠⠃⠀⠀⠈⢰⠀⠀⡇⠀⠀⠀⢅⠂⠀⠀⠆⠰⡄⠐⢶⣤⡳⣦⣪⢾⠿⣿⣿⣷⡿
⠀⠀⠀⠀⠀⡿⠁⠀⢀⣴⠋⡰⠁⠀⡴⠁⠀⠀⠀⠏⠐⠀⠀⠄⢸⠀⠀⡅⠀⠀⠀⢈⠀⠀⠀⠘⡄⠁⠀⠈⠻⣧⣬⡇⣀⠀⠉⠿⢯
⠀⠀⠀⠀⠀⠁⡀⣰⡿⢡⡮⠀⠀⣼⡓⠀⠀⠀⣸⠀⠀⠀⠀⡄⣾⠀⠀⢘⠀⡀⠀⠸⣿⣇⡀⠀⠐⡀⡐⠀⠀⠙⢧⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠙⢷⠏⠀⢀⣾⠏⢀⠂⠀⢰⠃⡀⠀⠀⢰⠃⡞⠀⠀⢀⡀⢹⠀⠀⢱⣷⡀⠀⠀⠹⣾⡄⠀⠀⠈⠻⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⣤⠎⠀⢠⣟⢠⠁⠀⠀⣾⠀⡇⠀⠀⠀⣧⠈⡀⠀⠐⣿⣷⠀⠀⠀⢸⣿⣦⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⡀⢀⣾⡇⡜⠀⠀⢰⣿⢰⡇⠀⠀⢠⣿⡀⠃⠀⠀⠨⢻⠷⡀⠀⠀⠹⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠠⡀⡸⠧⢻⠃⠀⠀⢸⣿⢧⣇⠀⠀⠀⠇⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠢⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠊⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⡁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⡀⡄⢀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⢧⡄⠀⠀⠀⠀⠀⠀⠀⣠⠖⠘⠉⠘⠋⠉⠙⠀⠶⠀⠈⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡆
⠀⠀⠀⠀⠀⠀⠁⠀⣀⣠⣄⢘⠖⣈⠁⢠⠶⠖⠒⠒⢒⠦⡀⠀⠙⠓⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⢂⡁⠋⠟⠧⠀⠒⠓⠀⠀⠈⠀⠀⠠⡶⠀⠀⠘⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠞⢣
⠀⠀⠀⡆⠒⠀⠉⠀⣀⠄⠘⣷⡄⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣈⡧⠀⠀⡢⠀⠀⠀⠀⠈⢂⠀⣀⢠⡀⠑⠋⠄
⠀⠀⠈⢧⣬⡦⠄⡚⠀⢸⣄⡈⠈⠀⠖⠀⠠⣄⡀⠙⡀⠀⠀⠐⠋⡀⣠⡆⠀⠐⡀⠀⠀⠀⠀⡼⢚⠵⣻⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢻⣧⠘⠁⠀⠺⡏⣱⣖⠀⠀⠀⡀⠀⢙⡆⠸⡇⢸⡄⠀⠉⠈⠁⠀⢰⣠⠀⠀⠈⢈⠰⠁⠀⡏⡀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⣿⣟⡙⡀⠀⠁⠘⠤⣹⡿⣴⠇⠘⠁⠀⠀⢤⣶⣄⡼⠞⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡄⢱⠀⡄
⠀⠀⠀⠀⠀⠋⠙⢀⣔⢁⠈⠉⠻⣾⣳⢶⠶⠞⣻⣃⣼⠔⠀⠠⠤⠀⠐⠛⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢠⣾⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⢮⣷⠏⠀⠀⠀⠙⠛⠛⠳⠿⠟⠉⠀⣀⣠⡀⢀⡀⠀⣘⠁⠀⠀⠀⠀⡀⡄⡇⣤⠄⣿⣾⣼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠅⢀⠀⠀⠀⢄⡀⠀⠀⠀⢂⣬⣀⡀⠘⢉⠴⠿⠭⠼⠭⠽⠿⠛⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⣞⢃⠀⠀⠉⠉⣿⠄⠊

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡅⢠⡄⢤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣩⣤⣥⣬⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣧⣤⢤⣤⣤⢤⣤⡤⢤⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠄⠤⣤⡄⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⢀⠔⠀⡖⠲⣄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢼⣦⣤⣤⣤⣤⣤⣤⣤⣤⣾⠀⠀⠀⡖⣿⣦⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⣿⠀⠀⠀⠙⢦⣸⠁⡴⠋⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁⡇⠀⠀⠀⠀⠀⠀⠙⠓⠋⠀⣀⡀⠀⠣⣀⠼⠀⠀⠀⢿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠼⠀⠀⠀⠀⠀⠀⢠⢖⣒⢢⡀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡇⠀⠠⠤⡆⣦⢴⠠⠰⢤⠀⠀⢀⣄⣴⠶⣶⣠⣄⢻⡘⠿⣡⠇⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⡉⢻⣤⣟⢋⣹⣇⠈⠉⠁⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⡿⣿⢤⣽⣿⣿⣿⣧⣤⣤⣤⣼⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠒⠛⠓⠀⠀⠀⠐⠚⠚⠚⠛⠓⠛⠛⠛⠓⠛⠛⠛⠛⠚⠛⠛⠓⠂⠀⠀⠀⠚⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣲⣶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⢠⣤⣤⣤⣤⡤⠀⠀⢸⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⢠⣄⠀⠀⡷⢺⡆⢀⣀⣀⣀⣀⣀⠀⠀⢸⣿⠀⢀⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣇⠀⠘⠋⠀⢀⡷⢼⠇⠈⠉⠉⠉⠉⠁⠀⠀⢘⣿⢠⢯⣙⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠘⠛⠛⠛⠛⠓⠀⠀⢸⡿⣣⢫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⢀⡜⡵⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢠⢎⠞⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢣⢏⢾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣝⡱⣣⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣾⣉⠟⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠠⠟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠋

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠘⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠐⢠⠀⠲⢀⣦⢰⣲⢠⠀⣖⠀⠠⡂⢰⠂⣖⢤⠒⢆⣠⠄⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠊⠀⠈⠈⠈⠚⠘⠘⠀⠓⠀⠀⠊⠘⠀⠃⠋⠂⠀⠀⠁⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠳⠦
⠀⠀⠀⡴⠉⠀⢴⠄⠀⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠈⠀⠀⠀⠸⡄⠛⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠒⠂⢒⠚⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣒⣒⣒⣦⣀
⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢀⠀⣀⠈⡇
⠀⠀⠀⠑⢤⡀⠀⠀⠀⡠⠋⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠠⠤⠤⠤⠤⠤⠤⠀⠒⠒⠒⠒⠒⠒⠂⠀⠤⠤⠤⣄⠀⠀⠀⡇⠀⠂⠀⠀⠠⠅⠀⠂
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡠⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠇⢀⣁⣈⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⢦⠀⠀⠸⠀⢀⠀⠐⠀⠀⠀⠀⠀⢠
⠀⠀⣀⡀⣀⣀⣠⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠀⠀⢀⢤⡀⡰⣤⠀⣤⢶⢀⣴⠀⠏⢡⠜⡷⠋⠀⠀⢠⠀⠀⡄⠈⠀⠉⠀⠀⠀⡀
⠀⡧⠈⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⠠⢈⠀⠀⠀⣦⠌⠀⡾⢁⣎⣰⠇⠰⠤⠊⠃⠘⠉⠀⠀⠀⠀⢀⠖⠁⠀⠀⢸⠀⠀⠑⠠⠤⠶⠤⠤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢰⡂⠀⠈⠀⠳⠠⠊⠑⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠴⠮⠀⠀⠀⠀⢸
⠀⠈⠀⠒⠒⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠡⠀⠀⠀⢸⣀⡀⠤⠀⣄
⠀⠀⠀⠀⠀⠀⢰⢧⢀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠔⠀⠀⠀⢸⠋⡠⠀⡆⠡⡱⠄
⠀⠀⠀⠀⠀⠀⠀⢸⠈⠁⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠶⠆⠠⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠂⠄⠀⢀⠓⠏
⠀⠀⠀⠀⠀⢠⠞⠋⠉⢳⠹⠄⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣈⣀⣀⣈⣴⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⡔⠈⠂⠤⠄⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠠⠤⠤⠤⠰⠒⠒⠠⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢆⣀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡩⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⡀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⠒⠀⠈⠙⢢⣀⠄⢀⣀⠀⠀⠘⠢⠤⠤⠤⠚⢀⣀⣀
⠀⠀⣠⠞⠉⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⣈⣉⡉⡩⠭⠛⠋⠢⠀⠀⢀⠜⠁⠀⠀⠀⢠⠊⡤⢀⠀⠀⠈⣲⡀⠀⠀⠀⢰⠁⠀⠀⠀⡿⢝
⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⡔⢘⠈⡡⠫⡃⠈⡀⠄⠀⣐⠀⡂⠀⣆⠀⠀⠀⢀⠔⠁⠮⠀⠀⠑⢀⠔⠡⠋⠀⠀⠀⢸⠀⠀⠀⠀⡧⠘
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡄⠀⡇⣂⠊⢀⠄⠀⢁⡁⠐⠠⠅⠲⠁⢄⠛⠒⠒⠒⢻⢇⡒⠤⢀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠰⠁⠁
⠀⠹⡄⠀⠀⢀⣀⠀⠀⡏⠉⠈⠇⠀⠐⠀⠁⠀⠈⠁⠀⠀⢀⠔⠁⡠⠔⢂⠑⡀⠀⠀⠀⡐⠉⠁⢀⠐⠚⢳⠀⠀⡠⢆⣠⡏⠀⠙⠒⠚⠃
⠀⠀⠈⢦⣰⠉⡀⠱⡀⠘⡈⣀⡄⠀⠀⠀⠀⢀⠤⠒⠀⠀⠙⠄⠈⠔⡨⠐⢀⠐⡄⠀⠀⡑⠠⠀⢀⠕⢀⠎⠀⠌⠀⠀⠉⠁⠊
⠀⠀⠀⠀⠙⢄⠈⢄⢠⠀⠑⠉⠈⠂⠢⡀⠀⢃⠀⠀⠀⠀⠀⠈⢀⠈⠔⠈⢀⡤⠊⠀⠎⠁⠀⠀⠁⡔⠁⠀⠀⢆⡀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⢢⠀⠀⠀⠀⠀⠈⢆⡀⠂⠁⠀⠀⡜⣫⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠡⠀⢀⠠⠐⠀⠉⠀⠀⠀⠀⢀⣡⠏⠐⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⣴⡄⣠⣤⣤⣌⣭⣍⣩⡁⠀⠀⠀⠀⢸⣿⢣⣿⡿⣱⣿⡜⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣰⠖⠒⠒⠒⠀⠀⠀⠀⠀⠀⣼⣿⣷⣿⣷⣿⣿⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣠⣴⣶⣶⣶⣶⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⣻⣟⣛⣛⡋⠄⠀⠀⠀⠀⠀⠀⠀⢠⣴⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠠⣴⣿⢻⣽⣿⢗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠿⠃⠀⠀⠀⢰⡟⢛⣦⠻⠓⠈⠁⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡐⣷⣿⢀⣀⡀⠀⠀⠘⠻⢿⣋⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⠾⠶⠿⠿⠿⠆⠀⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⢸⣿⣿⡇⢤
⠀⠀⠀⠀⠀⠀⠠⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⠀⠀⢸⣿⣿⡇⠼
⠀⠀⠀⠀⠀⠀⠠⠭⠬⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠤⠥⠤⠤⠬⠭⠭⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠖⠓
⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⡀⠉⠁⠀⠀⠀⣠⠰⠒⠋⠉⠹⠱⢢⢄
⠀⠀⠀⠀⡀⠀⠷⠇⠉⡆⠀⠀⠀⠀⠀⠀⠀⠠⠖⠒⠂⠈⠀⠀⠀⠀⠀⡼⡙⠄⠀⠃⠀⠀⠀⠂⠜⣡
⠀⠀⠀⠀⠀⠤⡀⠠⠚⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠕⠀⠀⠀⠀⠀⠀⠀⠈⠐⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠓⢄⣂⠀⢀⣀⣀⠀⣀⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⢥⣴⡼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⣴⣷⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣷⣶⣶⣤⣀⡀
⠀⣸⣥⣈⠙⠢⣉⠛⠦⢤⣄⣀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⢻⣿⣿⣿⣽⡯⣺⡿⣿⣹⣿⣿⣿⣶⣤⡀
⢀⣯⣄⡈⠑⠂⠈⠑⠀⠀⠈⠙⢿⣦⣀⢀⣼⣿⣿⡿⡿⢯⢹⠁⠀⠘⢸⣿⣿⡿⣡⢿⠙⠹⣿⣿⣿⣿⣦⡀
⢸⢧⢄⡁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠳⠁⠁⠀⠘⠀⠀⠀⡾⠉⣿⡿⠁⠁⠀⣈⣤⣿⣿⣳⢗⣿⣆
⡾⠶⢤⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣽⣦⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⣤⣿⣿⣿⠩⠶⡊⠤⠚⣧
⡿⠶⠶⢤⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡟⠡⠀⠀⣀⣲⡿⢒⣶⣤⡄⠀⠀⠀⠀⢀⣸⡿⠀⠁⠀⠚⣁⠟⠃
⣿⠛⠂⠐⠀⠈⠀⠐⠀⠒⠦⠦⣤⣄⡄⣀⣈⣻⠋⡯⠀⣿⡟⢁⡤⠤⣄⠉⢷⢠⡀⠔⠈⠉⠀⠀⠀⠀⠠⠌⠲⣛⣻
⣿⠟⠛⠒⠒⠀⠤⠀⠀⠀⠀⠀⠀⠋⠀⠉⠉⠋⠂⣧⠀⢽⣧⠸⢧⠶⣸⡇⠈⡏⡧⠀⠀⠀⠀⠰⣶⡶⠦⠤⠤⠼⠿
⢿⡛⣀⣉⡀⣁⠤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⢹⣆⠈⠛⠿⡶⠶⠿⠁⡼⠐⠀⠀⠀⠀⠀⠚⠻⢷⠠⢀⣐⢒⣚
⠸⣟⣋⣥⠠⠠⠐⠀⢀⡀⠥⠀⠀⠀⠀⠀⠀⠀⢠⡈⢉⣷⠶⠤⣤⡤⢴⣋⠼⠧⣀⠀⠀⠀⠀⠀⠀⠀⢉⠒⠨⣝⣻
⠀⢻⣯⣤⠴⠒⠁⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊⠉⠠⠀⠆⠀⠈⠉⠀⠀⠈⠻⣶⣄⡀⠀⠠⢀⠀⢌⡲⢿⡿
⠀⠀⢻⣧⠔⠂⢡⠔⢚⣡⠔⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⣿⡿⣿⠾⣕⠢⣛⠾⣽⣿⠃
⠀⠀⠀⢻⣖⣡⡴⠚⣉⡤⢒⣁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⡁⠀⠳⣌⠳⣍⢷⣼⠏
⠀⠀⠀⠀⠙⣧⡴⢚⣥⠖⣋⠕⢊⠄⡀⢠⠞⠀⠀⢀⠀⡆⠀⠀⢀⣾⣇⠀⠀⠀⠀⠀⠠⡀⠀⠀⢈⢷⣬⣿⠃
⠀⠀⠀⠀⠀⠈⠻⣿⣴⢎⣡⠞⡡⢊⡴⢃⠔⠄⢠⡏⡀⡇⢀⠀⣿⣿⣟⠀⡄⠂⠱⠀⠀⠱⡄⠲⣌⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣵⣞⣰⣏⣴⠏⡴⢠⡟⣰⢃⠂⡒⢸⡗⡟⢻⢀⠀⠸⣆⢣⡸⣦⢿⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣇⣾⣠⡏⢰⡇⢠⠀⡇⢨⡇⡇⢸⡇⣆⣷⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠿⠧⣿⣼⣷⣾⣷⣿⠾⠷⠾⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢦⡀⠀⠀⢀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⡄⠀⠀⠀⢙⣦⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡣⡋⡄⠀⠀⠊⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠅⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠢⠰⠆⠀⠀⠀⠀⠐⠠⠿⠐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠒⠀⠀⢀⡀⢀⣀⣒⠀⠀⠀⠤⠀⠐⣀⢤⣤⠷⠟⠍⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠉⠓⠓⠶⢤⣠⠀⢀⠀⠀⠤⣝⡈⠛⠛⠠⣀⣀⣤⡶⠿⠛⠉⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⠀⠈⠑⠓⠒⠲⡘⠀⡼⠟⠛⠋⠁⠀⢂⠨⠄⠁⠢⠀⡠⠞⠰
⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠄⠒⣀⠄⠀⠀⠀⠀⠀⠃⠀⠐⠀⠀⠀⠀⢈⠁⠁⠶⢤⡄⠜⢸⣏⡀
⠀⠀⠀⠀⠀⡬⠐⢧⡀⢠⢊⠰⣢⠀⠀⢠⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⣆⡀⢀⢄⠀⠸⡧⠆
⠀⠀⠀⠀⠀⠡⡈⣰⣿⠀⠀⠀⢂⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁⠓⠂⠀⠀⠳⡆
⠀⠀⠀⠀⠀⠀⠈⠚⠛⠀⠀⡤⠀⠀⠈⠫⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠄⠛⣙⢰⡆⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⢿⠂⠀⠀⠀⠀⠀⠀⠀⠘⠉⢉⣠⠖⠦⠉⠁⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠈⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠠⠘⠀⠁⠀⠀⢀⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣀⠀⠀⠀⠀⠀⠀⡎⢰⠄⡆⠀⠀⠀⠀⡠⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠒⠢⠤⠄⡹⢍⠄⠠⠖⠀⠉⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣹⣿⣉⣿⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⣿⣿⠿⠿⢿⣿⡿⠿⢿⡿⠿⢿⣿⡿⠿⠿⣿⣿⡿⠿⠿⣿⣿⡿⠿⠿⣿⣿⡿⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⣿⡇⢰⣶⠀⣿⣷⠀⢸⣷⡄⢸⣿⠀⣶⡆⢸⣿⠀⣶⡄⢸⣿⠀⣶⡆⢸⣿⠀⣴⡆⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⡇⠘⠟⠀⣿⣿⠀⢸⣿⡇⢸⣿⠀⠻⠃⢸⣿⠀⠻⠃⢸⣿⠀⠻⠃⢸⣿⠀⠻⠃⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⣶⣶⣾⣿⣿⣷⣿⣿⣷⣾⣿⣷⣶⣶⣿⣿⣷⣶⣶⣿⣿⣷⣶⣶⣿⣿⣷⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢉⣉⠙⣿⣏⠉⣿⡟⠉⣉⠉⣿⡟⠉⣉⠉⢿⣏⠉⢹⣿⠉⣉⠉⢻⣿⠉⣉⡉⢻⣿⡉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⠀⣿⣿⠀⣿⡇⢸⣿⠀⣿⡇⢘⣿⠀⢸⣿⡃⢸⣿⠀⣿⡇⢸⣿⠀⣿⡇⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣀⣀⣤⣿⣿⣀⣿⣧⣄⣀⣠⣿⣷⣄⣀⣠⣾⣿⣅⣸⣿⣄⣀⣠⣼⣿⣄⣀⣠⣼⣿⣇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⣿⡟⠋⠉⠙⣿⡏⠉⢻⣟⠉⢹⣿⡟⠋⠉⠙⣿⡏⠉⢻⡿⠋⠉⠙⢿⣟⠉⢹⣿⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⡇⢸⣿⠀⣿⣿⠀⢸⣿⣷⣾⣿⣷⣾⣿⣶⣿⣿⣶⣾⡇⠀⣿⠀⢸⣿⡇⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣀⣿⣧⣈⣉⣀⣿⣿⣀⣸⡟⢁⣤⣤⣤⣤⣤⣤⣤⣤⡈⢻⣧⣀⣉⣀⣼⣿⣇⣸⣿⣇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠛⠻⣿⡿⠛⠛⠻⣿⣿⡇⢸⣿⣿⡆⠉⠉⣿⣿⣿⡇⢸⣿⡟⠛⣿⡟⠛⢻⡿⠛⠛⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢰⣷⠀⣿⡇⢰⣷⠀⣿⣿⣧⡈⠙⠛⣿⣴⣄⣸⡟⠋⢁⣼⣿⣷⠀⣿⣿⠀⢸⡇⢰⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠈⠋⢀⣿⣇⠈⠋⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠀⢸⣇⠈⠛⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⢻⣿⣿⣿⡿⠛⢿⡿⠛⢿⣿⣿⣿⡟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠁⠀⢸⡇⠀⠈⠉⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡔⠉⢱⠚⡉⠈⡗⡞⠉⢱⢠⠆⠠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡴⡇⠈⡇⢹⠀⢸⠀⡇⠀⡇⡇⠀⡜⢸⠀⢰⢩⠃⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⣏⠆⢰⢸⠀⠸⡈⡆⠈⡆⡇⢠⠁⡇⠀⠃⡆⠀⡆⡜⠀⡞⡞⠈⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡈⢦⠘⡜⡄⢃⢇⠀⢃⢱⠀⡇⡇⢸⠀⡇⢸⢰⠁⢰⢡⠃⡜⡸⠀⡜⡰⢡⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⣄⠳⡌⢧⠘⡽⡈⡌⡄⠸⡄⠇⢡⢀⢸⢸⠀⡌⠸⠀⡎⡜⢰⢡⠃⡰⠱⢁⢎⠞⢀
⠀⠀⠀⠀⠀⠀⠀⡇⠘⢟⢆⠙⢄⢣⠘⡷⠘⡸⡀⢃⢸⢸⢸⢸⢸⠀⠇⡆⢰⢠⢃⠇⠆⣰⢡⢁⠎⠈⢠⢺⢄
⠀⠀⠀⠀⠀⠀⢘⠙⢦⡈⢣⡱⡈⢢⢡⠱⠡⢱⢣⠸⡀⡈⠸⢸⡈⠰⢰⠀⠇⡌⠎⡘⢠⢣⢃⢎⠎⡰⡡⠃⡜⠃
⠀⠀⠀⠀⠀⠀⣼⠳⣄⠱⡄⠱⡜⢆⠣⠡⠡⢂⢛⠄⠁⠃⠀⠘⠀⠀⠈⢸⢠⠰⠰⢁⢃⢆⢌⠎⡜⡔⢁⠞⡠⢳
⠀⠀⠀⠀⠀⠀⣇⠙⢌⡳⡌⢦⡘⢎⢣⠑⣥⢳⡄⠀⢀⢀⠀⡀⠀⠁⠇⠆⢂⢡⠁⠎⠈⢈⢂⢎⠞⡰⣣⢊⡴⣻
⠀⠀⠀⠀⠀⠀⢫⠳⣄⠙⠪⡢⡙⢌⠣⠱⡘⡦⠰⠈⣆⠘⡆⡇⡈⠐⠈⡠⢀⠃⡘⠄⠀⡠⢣⢎⢞⠝⡵⢋⠔⡟
⠀⠀⠀⠀⠀⠀⠀⠳⣄⠑⢄⠈⠪⣦⡣⡑⡕⡘⢷⣁⢹⡄⣿⣧⠃⣌⠓⢀⠌⡐⠀⠀⠜⡰⡑⣱⡵⢊⠔⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡘⠦⡘⢗⠹⣜⢴⡸⣿⡘⣷⣹⣿⢀⡏⠀⡎⡐⣀⠜⢌⢌⡾⡪⡫⢔⡡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⡢⡑⢿⣮⢳⡿⣷⢿⣿⣿⣼⡇⣸⡁⣠⢏⣾⣿⣿⣫⡾⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⢵⣝⣧⣿⣿⣿⣿⣿⣾⣧⣟⣴⣯⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠤⠀⠀⠀⠙⢿⣿⣿⣿⡿⠋⢀⣀⣈⣀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⣿⣿⡋⠹⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⣾⣢⣹⣧⠤⢬⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⢸⠻⣥⠛⠓⠈⢯⣿⣤⣤⣄⡈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣝⠧⣾⣤⠀⠀⠀⡀⣀⠘⣿⠀⠉⠉⠓⠺⠻⢻⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣾⣿⣷⣾⡿⠩⠈⢻⣇⣤⠤⣀⣀⣀⠀⠀⠀⢁⣟⣷⠟
⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢀⣿⣛⣷⣶⣤⠤⡉⡛⠛⣷⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠋⠩⣿⣿⣿⣿⠿⠿⣿⣿⣿⡏⠉⠉⠉⢩⡓⣾⡙⢿⣤⢘⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠛⠫⠟⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠤⢼⠼⠷⠿⠷⠾⠿⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠀⠀⢰⡟⠀⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢠⣶⡟⠋⠀⠀⠀⢠⡿⠀⠀⠀⠉⠛⣶⣤⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⡀⠀⣾⠃⠀⠀⣠⡴⠟⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⠇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣿⣿⣇⣀⢀⣉⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣶⠶⠶⠾⠷⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠸⠶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠉⠀⠀⠉⢻⣿⣿⣇⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣷⣀⡀⠀⣀⣼⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⣿⣿⠏⠁⠈⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠻⡍⢉⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢦⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠏⢳⡄⢹⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢠⠏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣼⡃⢸⠁⡇⠠⠤⠤⠀⠠⠤⠤⠤⠤⠤⠤⠤⠀⠀⢸⠀⠀⢀⡟⢦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢸⡀⠇⠀⠶⠶⠶⠶⠶⠶⠆⠀⠶⠶⠦⠀⠀⢸⠀⢀⡞⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠶⠶⠶⠶⠀⠰⠶⠶⠶⠶⠖⠀⠀⢸⢀⡞⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠒⠂⠀⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⢸⡞⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠐⠒⠒⠒⠒⠒⠒⠒⠂⠀⠒⠒⠀⠀⡾⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠐⠂⠐⠂⠐⠂⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⡼⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣀⣐⣂⣐⣂⣐⣂⣘⣃⡀⠀⠀⢀⣀⣀⣀⠀⣾⠞⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡃⠀⢈⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⢰⠋⠀⢹⡄⠀⠀⠀⠀⠠⠤⠀⠤⠤⠤⠤⠤⠤⠤⠌⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⢦⡼⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣤⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠋




⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠠⣤⢠⡄⠀⢀⣀⡄⠀⡀⠀⠀⣀⡀
⠀⠀⠘⣾⣿⣻⢿⣼⣹⣟⡷⢮⣭⠁⠙⣶⣀⣀⣀⣠⣤⠄⣀
⠀⠀⠀⠘⠿⣿⣿⣾⡟⣹⣿⣿⡿⣋⣻⣿⢿⣻⣿⣟⠿⢷⣬⡀⢢⣄⣀⣠⡴⢊⣿⢦⣀
⠀⠀⠀⠀⠀⠻⣿⡿⢷⣶⣶⣿⣷⣯⣿⣭⣽⣿⣿⠿⣷⣤⡙⠻⣷⣼⣿⣷⣆⣸⠃⠘⠻⠚⠦⣄
⠀⠀⠀⠀⠀⠀⠈⣷⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣄⡙⢿⣿⣿⣏⡤⠀⠀⠀⠀⠄⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣧⣄⣀⠀⠀⠀⠀⠉⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⣀⠈⠙⢿⣿⣿⣿⣿⣿⣶⣎⠀⠀⠀⡈⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠻⣿⣿⣿⣿⣿⣿⡙⠻⠿⣿⣿⣷⣦⣤⣛⠿⣿⣿⣿⣿⣿⣶⣶⡊⢀⠠⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣭⣿⣿⣿⣿⣿⣿⣿⠿⢽⣿⣿⣿⣿⣥⣠⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠿⠿⢿⣽⣶⣿⣿⣿⡿⣷⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣷⣯⣟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣷⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⣷⠀⠀⠀⠀⣼⠋⠙⣧⠀⠀⠀⠀⣾⠉⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠁⠀⠀⠀⠀⠈⢻⡟⠁⠀⠀⠀⠀⠈⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⡆⠀⠀⠀⢸⡇⠀⠀⠀⢰⡟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠒⠒⠒⠒⠚⠓⠒⠒⠒⠚⠓⠒⠒⠒⠚⠛⠒⠒⠒⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣿⠛⢷⡀⢸⡛⢳⡄⢠⡟⠛⣧⢠⣞⠛⠋⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣿⠀⣸⠃⢸⡁⣸⡇⢸⡇⠀⣿⠀⠙⠛⣷⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠛⠛⠁⠀⠘⠛⠋⠀⠀⠙⠛⠁⠀⠛⠛⠁⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣈⣉⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣉⣉⣉⣁⣀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠻⣦⣤⣤⣤⣤⣤⣤⣴⠏⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢙⢠⢸⢠⢢⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠉⠈⠁

⠀⠀⠀⠀⢢⠟⣹⡖⢠⣀⣀
⠀⠀⠀⠀⠀⣶⠇⣀⣿⠋⢁⣈⡶⠒⠶⠒⠤⡀
⠀⠀⠀⠀⠀⠸⣶⣿⠓⣄⡠⠋⠀⣰⠖⠀⠀⢈⡦⠤⠄⠀⠠⠤⠀⠤⡀
⠀⠀⠀⠀⠀⠀⢹⣁⠀⡜⠁⠀⠐⠋⠀⠀⢠⠞⠀⡔⠀⣀⣠⠄⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠸⣿⡟⢬⣿⠋⠀⠀⠀⣩⠃⠀⣀⠁⠀⠀⠈⠀⠀⠀⠀⠨⣦⠤⢀⡀⢀⣠⡤⠖⠋⠀⠀⠈⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠙⢦⣀⠀⡴⠁⣠⠂⢀⠉⠁⠒⠀⠀⠀⠀⠀⠀⢹⡄⠀⠈⠁⠀⢀⣀⣀⣤⠔⠋⠉⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣍⡿⠀⡼⢁⣴⣅⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⡀⠀⠀⠀⣤⠤⠰⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢏⣐⢶⣶⡏⢀⠞⢁⠿⣿⡟⠀⠀⠀⠀⠀⠀⢀⠞⠁⢀⣀⣀⣐⣀⣁⣠⣍⣤⣤⣤⣶⣶⣦⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⡟⣴⢻⣲⣦⠞⠺⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣄⢠⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⡷⡅⠈⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣛⣭⣵⣶⣿⣿⣿⡻⣿⣿⣧⡂⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣞⣞⣋⣩⣿⣟⠛⢻⣟⣿⡀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣾⣿⣿⣿⣿⣶⣶⣿⢮⣴⣦⣽⣷⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⠙⠷⣄⡠⢂⠀⠀⠀⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣦⣿⣌⠙⢻⣿⣿⡆⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⠀⣠⣀⠀⠉⠳⢀⠀⠀⠀⠀⠈⠁⠐⠒⠂⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣷⣾⢽⣃⠢⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⡷⢄⡀⠀⠁⠐⡢⠤⣀⠀⠠⠤⠤⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠛⢻⣿⣶⣿⣷⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣟⣠⡄⠀⣀⣀⣀⠴⠴⢶⣶⣶⣴⡦⣄⠀⠀⠱⣄⣀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉⠁⠀⠀⠀⠀⠉⠛⠿⣧⣤⣂⠀⠀⠉⣷⡀⠀⠀⠹⣿⣿⣿⡿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢯⣭⣏⣉⣭⣷⡀⠀⠀⢻⣿⣿⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣟⡿⣦⠀⠀⢿⣿⣿⣧⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣍⢐⣿⣷⣄⠀⠻⣟⣿⡿⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠝⢫⣿⣷⣦⣌⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉